Arizona Gov. Doug Ducey has steadfastly refused to discuss a replacement for Sen. John McCain until after the senator’s funeral, but political consultants say the governor faces an “unenviable” task when he does name an appointee.

McCain, who died Saturday after a yearlong battle with brain cancer, was described by political experts this weekend as “Arizona’s greatest face” and a “remarkably iconic figure.” While several names have been talked about in political circles, analysts say they all have big shoes to fill.

“John McCain changes the rules on an appointment because of who he was,” political consultant Jason Rose said. “The governor … needs to appoint someone with the seriousness and stature to respect the legacy of Senator McCain.”

And it’s not just McCain, another expert said, but a line of lawmakers who have played outsized roles in the Senate.

Under Arizona law, if there is a vacancy in one of the state’s two Senate seats, the governor is required to name a replacement of the same party as the senator who held the seat previously.

If the vacancy occurs more than 150 days before the primary, the seat comes up for election in the next general election. For a vacancy that occurs less than 150 days before the primary, the replacement senator holds the seat until the second general election after the appointment.

With Arizona’s primary this Tuesday, that means whoever is appointed to take McCain’s seat will hold it until 2020, when there will be an election to fill out the remaining two years in the current term. And the new senator must be a Republican.

After that, it’s anybody’s guess. And Ducey’s not talking.

The governor’s office said Sunday evening that there will be no announcement of a replacement until after McCain’s burial at his alma mater, the U.S. Naval Academy.

“Out of respect for the life and legacy of Senator John McCain and his family, Governor Ducey will not be making any announcements about an appointment until after the Senator is laid to rest,” said Daniel Ruiz II, senior adviser to Ducey. “Now is a time for remembering and honoring a consequential life well lived.”

Ducey also declined to discuss potential replacements while the senator was still alive. When news reports in May suggested that a meeting between Ducey and McCain and his family may have involved talks on a replacement, the governor’s spokesman at the time called such reports “disgraceful” and said Ducey would not “comment on any aspect” of such stories.

When Ducey does make a choice, it would be the first time in the state’s history that a Senate seat has been filled by the governor, according to a liston the Senate’s web page.

Despite Ducey’s refusal to discuss names, many have been mentioned among politics watchers.

Rose said McCain’s widow, Cindy, and his close friend and CEO of Plaza Cos., Sharon Harper, have been mentioned as possible appointees, along with former Sen. Jon Kyl and former Rep. Matt Salmon. Well-known business people, such as Barbara Barrett, Arizona Cardinals owner Michael Bidwill and former Phoenix Suns owner Jerry Colangelo, have also been discussed, Rose said.

Sean Noble, a partner in the lobbying firm Compass Strategies, said that any politician who comes next will be standing on the shoulders of some great Arizona men and women who preceded McCain.

“When you look at the string of Arizona senators and members of Congress that we’ve had, going from Carl Hayden to Barry Goldwater to Dennis Deconcini to McCain,” Noble said. “I mean, it’s a pretty remarkable record.”