Stringy Hodge-number conjecture for the Debarre dual fibration

Let $X\rightarrow\mathbb{P}^g$ be a Debarre system and let $\widehat{X}$ be its dual fibration, with fibres related by dual exact sequences of Jacobians and abelian varieties. The notation $h^{p,q}(X)$ denotes the Hodge numbers of $X$, while the stringy Hodge numbers of the possibly singular orbifold $\widehat{X}$ are the corresponding stringy invariants. Stringy Hodge-number conjecture for the Debarre dual fibration. The stringy Hodge numbers of $\widehat{X}$ equal the Hodge numbers of $X$. This is proposed as an analogue of the equality between the stringy Hodge numbers of the PGL Hitchin system and the Hodge numbers of the SL Hitchin system.