The Double Mirror conjecture for theta functions of Looijenga pairs

Let $(X,E)$ be a smooth Looijenga pair with affine interior $U=X\setminus E$, let $u:\mathcal T\to X$ be the universal torsor, and set $G=\mathcal T|_U$. Let $U^{\vee}$ and $V$ denote the relevant mirror constructions, with $V$ the mirror to $G$, and write $V^{\vee}$ for the dual mirror. Double Mirror conjecture. The double mirror satisfies

$$V^{\vee}\coloneqq (U^{\vee})^{\vee}=U$$

up to coefficients. This is the first of three conjectures proposed as part of a scheme for producing a theta-function basis of $\mathcal O(\mathcal T)=\operatorname{Cox}(X)$; the source gives no evidence resolving it.