Genus-two Hankel-determinant sigma-function conjecture

Let $\Delta_n$ be the Hankel determinants whose moments are defined by the recurrence in the source. Let $\tilde{\mathcal C}$ be the quintic curve

$$\tilde{\mathcal C}: y^2=4x^5+\sum_{j=0}^3\tilde{c}_j x^j,$$

which is isomorphic to the relevant sextic curve $\mathcal C$, and let $\sigma(\mathbf z)$ be its genus-two Kleinian sigma function, with period lattice $\Lambda$. For $\mathbf z_0,\mathbf z\in\mathbb C^2\bmod\Lambda$, where

$$\mathbf z=2\int_{\infty}^{\tilde P_2}\left(\frac{\mathrm d x}{y},\frac{x\,\mathrm d x}{y}\right)^T,$$

with $\infty$ the unique point at infinity on $\tilde{\mathcal C}$ and $\tilde P_2$ corresponding to $\infty_2\in\mathcal C$, let $\hat a$ and $\hat b$ be non-zero constants. Genus-two sigma-function conjecture. The determinants are given by

$$\Delta_n=\hat a\hat b^n\frac{\sigma(\mathbf z_0+n\mathbf z)}{\sigma(\mathbf z)^{n^2}}.$$

A proof would follow from an analytic solution for the iterates of the associated map. The proposed formula would imply a Somos-8 relation, or a Somos-6 relation under a certain constraint on $\mathbf z$, but the source gives no proof.