The inequality of the Thurston norm

Let $M$ and $N$ be admissible $3$-manifolds, and let $f\colon M\to N$ be a map that is surjective on $\pi_1(N)$ and induces isomorphisms

$$f_*\colon H_n(M;{\mathbb Q})\to H_n(N;{\mathbb Q})$$

for every $n\geq 0$. For $\phi\in H^1(N;{\mathbb R})$, write $f^*\phi$ for its pullback. Inequality of the Thurston norm. One should have

$$x_M(f^*\phi)\geq x_N(\phi).$$

This conjecture asks whether the Thurston norm is monotone under such maps. The inequality is proved when $\pi_1(N)$ is residually locally indicable elementary amenable, in particular when $N$ is fibered; the general case remains open.