Let $G$ be a finitely generated group. Suppose $G$ has an aspherical presentation with a countably infinite generating set. Does $G$ have an aspherical presentation with a finite generating set?
Here aspherical means: The presentation complex is aspherical, i.e. the universal cover of the presentation complex is contractible.
In my motivating example, $G$ is a two-generated subgroup of a small cancellation group. An answer to the same question for combinatorially aspherical (CA), diagrammatically aspherical (DA), or singularly aspherical (SA) presentations in the sense of [I. M. Chiswell, D. J. Collins, J. Huebschmann, "Aspherical group presentations", Math. Z. 178 (1981), 1-36] would also be highly interesting to me.
