It would be hard to get out in front of some members of Congress in trash-talking their own herd. Yet on Wednesday, four Democratic and four Republican senators standing together in front of TV cameras had something unalloyed to say for themselves: “There may be hope for us yet,’’ Sen. Mark Warner (D-Va.) told reporters at a news conference announcing the introduction of a truly bipartisan campus sexual assault bill.

The most important thing the law would do is force universities to recalculate the cost of hiding a problem so widespread that in surveys, one woman in five says she’s been assaulted during college. Currently, schools that dutifully report such attacks to the Education Department, as they’re required to do under Title IX, wind up looking worse than schools whose officials skirt the rules and hope for the best.

Just how common are attempts to pretend assaults only happen on other campuses? Well, at the universities my ­18-year-old daughter and I visited in the past two years, I routinely asked the appropriate officials how many sexual-
assault reports they’d had in the previous year — and was repeatedly told their number was zero, even though some of those schools I knew had high-profile cases. Helpfully, officials at two schools did volunteer info about notorious cases on other nearby campuses, though.

Right now, the only stick the Education Department has to try to get schools to comply is the threat that the university could, theoretically, lose every cent of its federal funding.

That’s “like me telling my kids I’m never going to speak to them again” if they don’t shape up, said Sen. Claire McCaskill (D-Mo.), who held a recent series of roundtable discussions on campus sexual assault and surveyed 350 colleges on their current practices. Could any threat be emptier?

Under the proposed bill, the Education Department could impose fines of up to 1 percent of a school’s budget — which in the case of, say, Harvard University, would add up to a tidy $42 million.

The bill, which McCaskill said she’d like to see on the Senate calendar in September, also would require training for staff and regular surveys of students, and it would provide funding for a confidential adviser for each student who reports an assault.

Other co-sponsors of the bill include Sens. Marco Rubio (R-Fla.), Richard Blumenthal (D-Conn.), Dean Heller (R-Nev.), Kirsten Gillibrand (D-N.Y.) , Charles E. Grassley (R-Iowa), and Kelly Ayotte (R-N.H.)

Astonishingly, every one of them had something intelligent to say on the issue. They didn’t sound as though they’d been briefed by an aide on the way over.

“Sometimes the victim is treated worse than the person who has committed the crime,’’ said Grassley, while Rubio declared that there can no longer be any “special preference because someone can dunk a basketball or throw a ball 80 yards down the field.”

Ayotte noted that in 20 percent of the schools surveyed, “we have actually found evidence that athletic programs [themselves] have investigated sexual assaults.”

Annie Clark, a 2011 graduate of the University of North Carolina and co-founder of an advocacy group called End Rape on Campus, said that after she reported being raped at UNC, she was told by a university staffer that “rape is like a football game; you should look back on it and think about what you could have done differently.’ ’’

Her own message to others who have been assaulted is a little different: “You’re not alone, it’s not your fault, and we believe you.”

A young woman named Anna, who was the subject of a recent story in the New York Times and who asked to be identified only by her first name, said she had reported being attacked by multiple football players early in her freshman year at Hobart and William Smith Colleges in New York.

“I stand here today, and I’m not okay,” she said at the news conference. Though a nurse examiner found evidence of blunt-force trauma, the district attorney closed the case a day after it was referred to him, and the student was found not responsible at a campus disciplinary hearing.

Anna’s mother, Susan, said in a tremulous voice how proud she was of her daughter and of others who have reported assaults despite enormous pressure to keep quiet. She asked senators to fight to get the bill onto the floor “also for those who didn’t make it, for Jeanne Clery’s parents and Lizzy Seeberg’s parents and many, many more.’’ Clery was raped and murdered in her dorm at Lehigh University in 1986, and Seeberg committed suicide in 2010, 10 days after reporting that she’d been sexually assaulted by a Notre Dame football player. Wednesday would have been her 23rd birthday.

Gillibrand, too, said she wanted to thank both those present and those “who aren’t here today.”

In concluding his remarks, on a day that also produced nearly unanimous passage in the House of a Department of Veterans Affairs reform package, Rubio said, “I pray this doesn’t get caught up in all the other things that go on in this city”— politics, he meant.

But Gillibrand said she actually didn’t think that would be a problem, and McCaskill said the only real obstacle will be “trying to elbow our way” onto the schedule in the fall.

After the cameras were turned off and most reporters had gone, McCaskill and Gillibrand hugged each of the five survivors. “You all kicked it,’’ McCaskill told them. “You just kicked it.”