The Berstein–Schwarz formula for the LS-category of a group homomorphism

Let $f:G\to H$ be a homomorphism between discrete groups. Let $Bf:BG\to BH$ be a map between Eilenberg–MacLane spaces inducing $f$ on fundamental groups, and let $\beta_H\in H^1(H;I(H))$ be the Berstein–Schwarz class of $H$, where $I(H)\subset\mathbb Z[H]$ is the augmentation ideal. The LS-category of $f$ is defined by $\operatorname{cat}(f)=\operatorname{cat}(Bf)$. Berstein–Schwarz conjecture. The LS-category of $f$ satisfies

$$\operatorname{cat}(f)=\max\{k\mid f^*\beta_H^k\neq 0\}.$$

This conjecture extends the Berstein–Schwarz characterization of the cohomological dimension of a group from the identity map to arbitrary homomorphisms. The supplied text does not state whether the formula is known or unresolved.