I only ask because I don't know how to look for the answer.
It's still an open problem. The isomorphism problem for hyperbolic groups (a much smaller class) was only solved recently by Dahmani and Guirardel (see here), following work of Sela.
In the same vein, the conjugacy problem is also still open for automatic groups. It has been solved for biautomatic groups, but it is also still open whether all automatic groups are biautomatic.