Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?
To offer a little more motivation, Roe proves that all hyperbolic groups have finite asymptotic dimension, a result that was particularly interesting at the time as it ensured that such groups satisfy the Novikov conjecture due to the work of Yu.
Small cancellation groups are a rich and interesting subclass of hyperbolic groups - surface groups (genus $\geq 2$) and random groups in the few relator or low density models are almost surely examples. Calculations and computations are often easier for these groups - for instance they have cohomological dimension at most 2. Unfortunately I cannot find an argument which says either that such groups have uniformly bounded asymptotic dimension, or proving that this is unbounded.