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On Tuesday, JPMorgan Chase shareholders rejected a proposal to split the roles of CEO and chairman, both held by the bank's iconic leader, Jamie Dimon. The measure received support from roughly 32 percent of shareholders, which is actually less than the 40 percent support a similar proposal saw last year.

Dimon, once seen as the sage of Wall Street, has come under fire after lapses in its chief investment office resulted in a loss of more than $6.2 billion. But how much would splitting the roles of CEO and chairman really matter? Leadership thinkers are, well, as split on the issue as shareholders are. Supporters of separating the role say it leads to better governance (the CEO, after all, is supposed to answer to the board), less costly CEOs and more independent boards.

Detractors, meanwhile, say splitting the job isn't always that effective, even though it often makes the transition from one CEO to another less disruptive. It can "escalate palace intrigue" and lead to distracting power struggles, wrote Yale professor of management Jeffrey Sonnenfeld in the New York Times recently. Many companies that split the two roles later recombined them. And recent research has shown that splitting the role at the behest of activist shareholders can actually lead to poorer performance down the line.

The best answer on how to navigate the chairman-CEO dilemma might be the same one that applies to most relationships—it's complicated. In another recent study, two researchers from Indiana University's Kelley School of Business found that low-performing companies tend to benefit from separating the two jobs, while high-performing companies suffer when the jobs are split.

That may be why Dimon still has both jobs. Despite big questions about Dimon's role in the "London Whale" trading scandal and potential changes to the make-up of JPMorgan's board, the bank still posted record profits last year even after posting huge losses. And JPMorgan's shares have risen by more than 50 percent in the last 12 months.

Jena McGregor is a columnist for On Leadership.

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Jena McGregor writes a daily column analyzing leadership in the news for the Washington Post’s On Leadership section.