While this is pretty much too late to help OP, you don't have to be at the gate at boarding time. Just 10 minutes before departure (ie 5:45). Looks like the SFO flight is about to arrive (5:05) and the connection is C15-C28, so OP should be just fine.
To be precise and presuming that this is UA, it is T-30. Even for a domestic flight it is T-15.
T-10 is when the flight should be closed, but that is not the same as the deadline for passenger boarding as the difference is intended to permit offloading no shows and boarding standbys.
OP is unlikely to be offloaded until just before the gate is physically closed, but if the flight is overbooked and at risk of oversale, UA will offload a no show before paying IDB/VDB.