Beauville's generic vanishing conjecture for the Albanese morphism

Let $X$ be a projective manifold, let $\alpha:X\to \operatorname{Alb}(X)$ be an Albanese morphism, and let $\operatorname{Pic}^0(X)$ denote the group of topologically trivial line bundles on $X$. A property holds for generic $L\in\operatorname{Pic}^0(X)$ when it holds outside a proper analytic subset.

Beauville's conjecture. If

$$\dim\alpha(X)>1,$$

then

$$H^1(X,L)=0$$

for generic $L\in\operatorname{Pic}^0(X)$.

The source presents this as a conjecture related to the dimension of the paracanonical system and as a generic vanishing statement. Its resolution is not given in the supplied text.