Interesting. I'm definitely a Deer. I get the sense (could be 100% wrong) native CNYers, for the most part, are Bears, "outsiders" (myself included) are predominantly Deers.
"deer" for sure. Do not know where the rift originated between the two pronunciations (perhaps the soft 'Y' one is more mid-western in origin?). Anyways, I grew up on the East side, and I never heard it pronounced sarah-cuse, not once. I only started hearing it relatively recently when certain broadcasters used it while calling a game or on highlights. To me syr should be pronounced like syrup, or like the town in Italy that Syracuse is named for.
What do the announcers say? I think Mike Tirico may say seer. I would say Jay Bilas says sarah. Bill Raftery may just say the Cuse. Digger may also say seer, actually.
I'm also originally from Syracuse. Born there, grew up there, graduated high school there. I'm definitely a deer. I've only ever heard bears when watching the Orange on TV, as the majority of announcers pronounce it that way.