A very public battle is raging over how ExxonMobil should face questions about long-term oil demand and prepare for a carbon-constrained world.

Driving the news: Investment group Engine No. 1, which is nominating board members it says are equipped to deal with these dynamics, this week gained new support.

• Two huge pension funds — California Public Employees' Retirement System and the New York State Common Retirement Fund — have thrown their weight behind it.
• It adds to prior backing from the California State Teachers' Retirement System, so now, per Pensions & Investments, the country's three largest pension funds back the effort.

Why it matters: The unusually high-profile shareholder battle is something of a microcosm of larger questions about the future of Big Oil.

• But it's also specific to Exxon, one of the world's most powerful companies, which has not sought to diversify as widely as its European peers (though oil-and-gas remains the dominant business for all of them).

What they're saying: Engine No. 1, in an 81-page presentation posted this week, says Exxon has "significantly underperformed and has failed to adjust its strategy to enhance long-term value."

• The slide deck argues that Exxon has long lagged behind its Big Oil peers by multiple metrics of investor returns.

The other side: Exxon, which has seen significant share price recovery over the last six months, has been striking back against the claims that it's poorly positioned and underperforming.

• Exxon yesterday posted its own detailed slide deck laying out the case for why its strategy will provide long-term value.
• It's the latest of recent moves including new board additions and vowing new emissions intensity cuts.
• The presentation delves into its investments in carbon capture while arguing that Exxon is well-poised to deliver strong returns in oil-and-gas, which it notes will remain immense markets for decades despite low-carbon energy growth.
• And one of its new board members, Jeff Ubben, defended Exxon's climate strategy in an interview with the Financial Times.

What's next: It comes to a head at Exxon's May 26 shareholder meeting. Reuters reports that the hedge fund D.E. Shaw, which has pushed Exxon to change its approach, now intends to vote with the company.