Mirror line bundle conjecture for parabolic very stable upward flows

Let $G$ be a reductive group with Langlands dual $G^\vee$, let $C$ be the underlying curve, and let $D=\{c_i\}$ be the parabolic divisor. Let $(E,\varphi,Q)$ be a smooth fixed point of Borel type with twisted multiplicity divisor $w:=w(E,\varphi,Q)$ supported at $D$. Assume there is a universal parabolic $G^\vee$-bundle

$$\mathbb E\to\mathcal M_{sp}(G^\vee,\alpha)\times C$$

with universal reductions to $B^\vee$ at the points $c_i$, and write $\mathbb E_i$ for the resulting $B^\vee$-bundle over $\mathcal M_{sp}(G^\vee,\alpha)$. For each $c_i$, let $\mu_i^{-1}:=\mu_{c_i}(E,\varphi)^{-1}$ be the associated antidominant minuscule cocharacter, viewed as a character of $B^\vee$. The line bundle on $\mathcal M_{sp}(G^\vee,\alpha)$ mirror to the structure sheaf $\mathcal O_{W^+_{(E,\varphi,Q)}}$ on $\mathcal M_{sp}(G,\alpha)$ is given by

$$\mathcal L_w:=\bigotimes_{c_i\in D}\mu_i^{-1}(\mathbb E_i)$$

up to tensoring by a fixed line bundle, independent of $w$, determined by the chosen normalisation of $\mathbb E$. This proposes the Fourier--Mukai mirror of a very stable upward flow that is a section of the Hitchin map; the normalisation ambiguity is independent of the twisted multiplicity divisor, while the proposed correspondence itself remains conjectural.