Self-reciprocity conjecture for the partial Hasse polynomials of the double cover of $W_{17}$

Let $\Pi_{17}$ be the double cover of the Teichmüller curve $W_{17}\hookrightarrow\mathbb{H}^2/\mathrm{SL}(\mathcal{O}_{17}\oplus\mathcal{O}^{\vee}_{17})$. For $j\in\{1,2\}$ and every prime $p\ne 2,17$, let $\mathrm{ph}^{\Pi_{17}}_{p,j}(J)$ be the corresponding partial Hasse polynomial, and let $n_{p,j}$ be the degree of $\mathrm{ph}_{p,j}$. Self-reciprocity conjecture. The partial Hasse polynomials satisfy

$$\mathrm{ph}^{\Pi_{17}}_{p,j}(J)=\mathrm{ph}^{\Pi_{17}}_{p,j}(J^{-1})\cdot J^{n_{p,j}}.$$

The conjecture asserts reciprocal symmetry for the partial Hasse polynomials in the computed degree-two cover of $W_{17}$. The source reports that this property holds in the cases computed and conjectures that it holds in general, but gives no proof or resolution.