Grant's LS-category conjecture for group homomorphisms

Let $\phi:\Gamma\to\Lambda$ be a group homomorphism. The LS-category of the homomorphism, $\operatorname{cat}(\phi)$, is the minimum number $k$ such that a classifying-space map inducing $\phi$ can be covered by $k+1$ open sets on which its restrictions are nullhomotopic. The cohomological dimension $\operatorname{cd}(\phi)$ is the maximum $k$ for which there exists a $\mathbb{Z}\Lambda$-module $M$ such that the induced map

$$\phi^*:H^k(\Lambda,M)\to H^k(\Gamma,M)$$

is nonzero.

Grant's LS-category conjecture. For any group homomorphism $\phi:\Gamma\to\Lambda$,

$$\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$$

This conjecture extends the Eilenberg–Ganea equality from groups to homomorphisms. The inequality $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ is known in general, but the proposed equality is false: Tom Goodwillie gave a counterexample consisting of an epimorphism from an infinitely generated group to $\mathbb{Z}^2$ with $\operatorname{cd}(\phi)=1$.