Guardia's Mumford isomorphism conjecture for hyperelliptic period matrices

Let $X$ be a hyperelliptic Riemann surface of genus $g\geq 2$, let $\tau$ be a period matrix associated to $X$, and let

$$\{\eta_1,\ldots,\eta_g,\eta_{g+1},\ldots,\eta_{2g+2}\}\subset\mathcal{F}$$

be a fundamental system, where $\mathcal{F}$ is the set of fundamental systems obtained from permutations of the Weierstrass points of $X$. Write $J(\eta_1,\ldots,\eta_g)(\tau)$ for the quantity associated with the first $g$ theta characteristics, and let $\vartheta[\eta_k](0;\tau)$ denote the theta constant associated with $\eta_k$. Guardia's conjecture. The formula

$$J(\eta_1,\ldots,\eta_g)(\tau)=\pm\pi^g\prod_{k=g+1}^{2g+2}\vartheta[\eta_k](0;\tau)$$

holds. This conjectural identity is an explicit form of the Mumford isomorphism for hyperelliptic curves, relating the Jacobian-type expression $J$ to a product of theta constants. The source attributes it to Guardia and presents it as a conjecture; the notation $J$ and the omitted formula label should be checked against the surrounding definitions.