The elliptic stable-basis duality conjecture under symplectic duality

Let ${\mathfrak{X}}$ and ${\mathfrak{X}}^!$ be symplectic-dual varieties, let $p_1,p_2\in X^H$, and let $p_i^!$ denote the corresponding points in $X^!$. If ${\mathrm{ind}}_{p}$ and ${\mathrm{ind}}_{p^!}$ are the attracting parts of the chosen polarizations, define

$$\sigma_{{\mathfrak{X}},{\mathfrak{X}}^!}(p)=(-1)^{\operatorname{rk}{\mathrm{ind}}_p+\operatorname{rk}{\mathrm{ind}}_{p^!}}.$$

Elliptic stable-basis duality conjecture. For any $p_1,p_2\in X^H$,

$$\sigma_{{\mathfrak{X}},{\mathfrak{X}}^!}(p_1)\left.{\mathrm{Stab}}^{ell}_{{\mathfrak{X}}}(p_1)\right|_{p_2}=\sigma_{{\mathfrak{X}}^!,{\mathfrak{X}}}(p_2^!)\left.{\mathrm{Stab}}^{ell}_{{\mathfrak{X}}^!}(p_2^!)\right|_{p_1^!}.$$

This is presented as an expected duality of elliptic stable bases associated with symplectic duality; the supplied text does not establish it in general.