FORTUNE — With each passing day, it looks like things are only getting worse for the media empire that Rupert Murdoch built. And with mixed reports over whether Murdoch will step down as CEO (potentially leaving COO Chase Carey to take the reins) in the run-up to his testimony before parliament today, what will be this beleaguered media mogul’s fate?

In contrast to his larger than life image and the reality of the power he wields, last week in an email to Dow Jones employees, Murdoch wrote: “Let me emphasize one point — News Corporation is not Rupert Murdoch.”

Currently, Murdoch holds both the CEO and chair positions at News Corp (NWS) along with approximately 40% of the voting rights of the firm.

But given all that has occurred at News Corp in the recent years under his reign, should he resign as CEO or chair or both?

Clearly, the company needs to clean house and update its board. Boards that fail to look into serious allegations that drag over many years are not doing their job.

At News Corp, the issue is even more critical because of the control Murdoch exercises as a shareholder over voting rights at the firm. What are the checks on so much power and the rights of minority shareholders for protection?

The question of whether to separate the CEO and chair positions (so two, separate people occupy those posts) often arises when crisis hits. (Think Ken Lewis at Bank of America (BAC).) Under such a scenario, a new chair could be appointed — or Murdoch could vacate the CEO slot and become the chair.

But as a chair, Murdoch’s independence would be a problem.

John M. Nash, founder and president emeritus of the National Association of Corporate Directors (Full disclosure: Nash has done work for me and my firm), believes that boards should do more than just separate the chair and CEO positions. “Chairs should not be the former CEO,” as it weakens the governance structure, he says.

What about the CEO spot for Murdoch?

As a general matter of good governance, regardless of News Corp’s woes, Nash thinks that CEOs should not sit on any board, including their own company’s board.

The CEO can attend board meetings, but “shouldn’t have a vote,” Nash says.

He believes it is important to boardroom effectiveness to be clear about who’s the “hired gun”, who runs the meeting, and who is holding the CEO accountable (for his or her actions). CEOs often have a difficult time separating their role on the board from their role as a member of management.

When I’ve asked CEOs who don’t sit on their company’s board what they think about it, they tell me they don’t mind the arrangement at all. It works well for them and does bring clarity related to who is accountable to whom.

Examples of S&P 500 companies with CEOs who do not sit on their company’s board include O’Reilly Automotive, Titanium Metals Corporation, GameStop, Tyson, and Sara Lee.

Perhaps if CEO Murdoch gave up his board seat, a truly independent board could be installed. (Murdoch could stay on as CEO without a board vote.)

Then, that newly constituted board could go about the business of making decisions on critical items like CEO succession, conduct a proper internal investigation, and determine who stays and who goes.

Sure, it may not happen, but anything is possible as this saga unfolds.

Eleanor Bloxham is CEO of The Value Alliance and Corporate Governance Alliance (http://thevaluealliance.com), a board advisory firm.