Monotone invariant conjecture for non-virtually-trivial circle bundles

For each dimension $n$, let $I_n$ assign a numerical invariant to homotopy $n$-manifolds. The invariant is required to satisfy, for every map $f\colon M\longrightarrow N$, the degree inequality

$$I_n(M)\geq |\deg(f)|\cdot I_n(N).$$

Monotone invariant conjecture. In every dimension $n$, there is such a homotopy $n$-manifold numerical invariant $I_n$ that is positive and finite on every circle bundle over a closed aspherical manifold with hyperbolic fundamental group that is not virtually trivial.

The conjecture is motivated by the domination semi-norm and the Seifert volume, which provide monotone invariants in related settings. The source presents this as a suggestion, and no resolution is given.