Dominance conjecture for higher Albanese maps of special varieties

Let $X$ be a normal quasi-projective variety which is either weakly special or $h$-special. For every natural number $s$, let $\operatorname{alb}^s\colon X\to\operatorname{Alb}^s(X)$ denote the $s$-th higher Albanese map.

Dominance conjecture for higher Albanese maps. The map

$$\operatorname{alb}^s\colon X\to\operatorname{Alb}^s(X)$$

is dominant for every $s$.

This proposes a geometric strengthening of the known dominance of the classical Albanese map for weakly special and $h$-special varieties. Its general validity is open.