Mirror of the canonical Hecke-cycle Lagrangian

Let $C$ be the underlying curve, let $c\in C$, and let $W^+_0$ and $W^+_c$ be the upward flows from the canonical fixed point and from the fixed point indexed by $c$, respectively. Their union $W^+_0\cup W^+_c$ is a closed Lagrangian cycle. Hecke-cycle mirror conjecture. The mirror of the closed Lagrangian cycle $W^+_0\cup W^+_c$ is the universal adjoint bundle for $c$ in the $\operatorname{SO}_3$ moduli space. The preceding discussion identifies the union as a closed subspace whose symplectic form vanishes at smooth points and describes the associated Hecke curve; the asserted mirror identification remains open in the supplied text.