State treasurers from Illinois, Rhode Island and Pennsylvania, and New York City Comptroller Scott Stringer, co-filed the proposal. They oversee money including pension funds and joined activist and original filer Trillium Asset Management.
A similar shareholder proposal seeking an independent chair was defeated in 2017 at Facebook, where Zuckerberg's majority control makes outsider resolutions effectively symbolic.
Rhode Island State Treasurer Seth Magaziner said that the latest proposal was still worth filing as a way of drawing attention to Facebook's problems and how to solve them.
"This will allow us to force a conversation at the annual meeting, and from now until then in the court of public opinion," Magaziner said in a telephone interview.
A Facebook spokeswoman declined to comment.
At least three of the four public funds supported the 2017 resolution as well. The current proposal, meant for Facebook's annual shareholder meeting in May 2019, asks the board to create an independent board chair to improve oversight, a common practice at other companies.
It cites controversies that have hurt the reputation of the world's largest social media network, including the unauthorized sharing of user information, the proliferation of fake news, and foreign meddling in U.S. elections.
Illinois State Treasurer Michael Frerichs said in an interview that, while an independent chair might not have prevented all the issues, "there might have been fewer of these problems and less of a drop in share price" at the company.
Shares of Facebook have had a rocky year, under pressure from revelations about the privacy and operational issues as well as concerns over slowing revenue growth. They closed Wednesday at $159.42, 10 percent lower than at the start of the year and well off a closing high of $217.50 reached on July 25.
The 2017 resolution received the support of a slim majority of outside investors, according to the public fund leaders' calculations. Magaziner and Frerichs said they planned to talk with larger Facebook investors in coming months to seek their support.
Among funds that are Facebook's largest investors, the Vanguard Total Stock Market Index Fund and Fidelity Contrafund voted against the 2017 proposal, securities filings show, while the American Funds Growth Fund of America supported it.
American Funds representatives did not reply to requests for comment on Wednesday. Spokespeople for Fidelity and Vanguard declined to comment. Contrafund manager Will Danoff was supportive of Facebook's response to problems in an investor note in August.
In opposing the 2017 proposal, Facebook said an independent chair could "cause uncertainty, confusion, and inefficiency in board and management function and relations."
Zuckerberg has about 60 percent voting rights, according to a company filing in April.
The New York City Pension Funds owned about 4.5 million Facebook shares as of July 31, while Trillium held 53,000 shares.
The Pennsylvania Treasury held 38,737 shares and the Illinois Treasury owned 190,712 shares as of August. Rhode Island funds hold 168,230 Facebook shares, a spokesman said.