Equality of the LS-category and cohomological dimension of a homomorphism

Let $\phi:\Gamma\to\pi$ be a homomorphism of discrete groups. The LS-category $\operatorname{cat}\phi$ is the least integer $k$ such that a classifying-space map $B\Gamma\to B\pi$ inducing $\phi$ admits an open cover of $B\Gamma$ by $k+1$ sets on each of which the map is nullhomotopic. The cohomological dimension $\operatorname{cd}(\phi)$ is the maximum integer $k$ for which there is a $\pi$-module $M$ such that the induced map $\phi^*:H^k(\pi,M)\to H^k(\Gamma,M)$ is nonzero. Equality conjecture. For every group homomorphism $\phi:\Gamma\to\pi$, one has

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

The inequality $\operatorname{cd}(\phi)\leq\operatorname{cat}\phi$ is known from the Berstein–Schwarz class, and equality holds for the LS-category and cohomological dimension of a single discrete group. The conjecture asks whether this equality extends to all group homomorphisms.