Could there exist a finitely presented group with unsolvable conjugacy problem, in which it is decidable whether a word over the group generators is a shortest representative of an element in its conjugacy class?
Here is some motivation. For many of the known families of automatic groups, such as hyperbolic groups, virtually abelian groups, Coxeter groups, Garside groups, Artin groups of large type, it can be shown that the set of all geodesic words (at least over an appropriate generating set) is a regular set. Conversely, if you assume that the geodesics form a regular set, then you cannot (apparently) deduce anything much at all about the group, beyond the fact that it has solvable word problem. But in fact the set of geodesic words being recursively enumerable is already enough to imply solvable word problem.
Moving on to the conjugacy problem, it seems that in at least some of these classes of groups, such as hyperbolic groups and Artin groups of extra-large type (not sure yet about large type), the set of all words that are shortest representatives of elements in their conjugacy class forms a regular set. But in that case, the converse hypothesis (regular or perhaps recursive set of shortest words in the conjugacy class) does not appear to imply solvable conjugacy problem.
