Effective characterization of semi-abelian varieties

Let $d \geq 1$ be an integer. For a smooth complex variety $V$ of dimension $d$, let $\overline{q}(V)$ denote its logarithmic irregularity, let $\overline{P}_m(V)$ denote its $m$-th logarithmic plurigenus, and let $a_V \colon V \to A(V)$ be its Albanese morphism to the universal semi-abelian variety.

Effective semi-abelian characterization conjecture. There exists an integer $k = k(d) \geq 1$ such that, whenever

$$\overline{q}(V) = \dim(V) \quad\text{and}\quad \overline{P}_1(V) = \dots = \overline{P}_k(V) = 1,$$

then $a_V \colon V \to A(V)$ is an isomorphism away from a closed subset of $A(V)$ of codimension at least $2$.

This conjecture seeks an effective quasi-projective analogue of the Chen–Hacon characterization of varieties birational to abelian varieties, replacing projective irregularity and plurigenera by logarithmic invariants and describing the resulting semi-abelian Albanese morphism in codimension one. The supplied source gives no evidence of resolution.