But meanwhile, to the north and south, they have made their most rapid gains for six months in the surrounding area, seizing swathes of territory from Russian troops.
"Wagner troops climbed into Bakhmut like rats into a mousetrap," Oleksander Syrskyi, commander of Ukraine's ground forces, told troops at the Bakhmut front this week.
The battle for Bakhmut has revealed a deepening split between Wagner, a mercenary force that has recruited thousands of convicts from Russian prisons, and the regular Russian military.
For two weeks, Prigozhin has been issuing daily video and audio messages denouncing Russia's military leadership, often in expletive-laden rants.
Moscow has long claimed that capturing Bakhmut would be a stepping stone towards advancing deeper into the Donbas region it claims to have annexed from Ukraine.