Kanev–Lange polarization-type conjecture for simply ramified $B_n$-coverings

Let $p:C\overset{\pi}{\longrightarrow}C'\overset{g}{\longrightarrow}\mathbb{P}^1$ be a simply ramified $B_n$-covering with $C$ irreducible. Let $\mathfrak{D}_s$ and $\mathfrak{D}_{\ell}$ be the branch loci corresponding respectively to short and long reflections, let $P(X,\delta)$ be the associated Prym–Tyurin variety, and let $\hat P'$ be the dual of the Prym variety $P(C,C')$. Kanev–Lange's polarization-type conjecture. The homomorphism $\mu:\hat P'\to P(X,\delta)$ is an isomorphism. Equivalently, the polarization type of $\Theta_{JX}|_{P(X,\delta)}$ is $(2^{n-2},\ldots,2^{n-2})$ if $|\mathfrak{D}_s|=0$ or $|\mathfrak{D}_s|=2$, and is $(2^{n-2},\ldots,2^{n-2},2^{n-1},\ldots,2^{n-1})$ if $|\mathfrak{D}_s|>2$, with $2^{n-2}$ appearing $\frac{1}{2}|\mathfrak{D}_{\ell}|+1-n$ times and $2^{n-1}$ appearing $\frac{1}{2}|\mathfrak{D}_s|-1$ times. The supplied status evidence says this conjecture remains open for simple $B_n$-coverings of $\mathbb{P}^1$ with $n\geq4$.