Surface-group algebra presentation conjecture via the noncommutative Weil representation

Let $C$ be a genus-$g$ Heegaard surface with fundamental group $\pi_1(C)$, let $X$ be the associated three-manifold, and let $L_+$ be one of the two lattices in the Heegaard splitting. Let $\mathbb C\langle\!\langle L_+\rangle\!\rangle$ be the completed noncommutative Laurent-polynomial algebra, let $\mathbb C\langle\!\langle L_+\rangle\!\rangle_{\operatorname{cyc}}$ be its cyclic quotient, and let $\wp$ be the conjectural projective mapping-class-group representation. For the mapping class $u$ satisfying $L_-=u(L_+)$, define $\Phi=\wp(u)(1)$. Surface-group algebra presentation conjecture. There is an algebra isomorphism

$$\mathbb C[\pi_1(X)]\cong\mathfrak A(F,\Phi),$$

where

$$F=\mathbb C\langle\!\langle L_+\rangle\!\rangle,\qquad \Phi=\wp(u)(1)\in\mathbb C\langle\!\langle L_+\rangle\!\rangle_{\operatorname{cyc}}.$$

This is part of the proposed trigonometric counterpart of the Heisenberg and Weil representations; the source gives no resolution evidence.