The quantum symplecticity conjecture for the canonical solution

Let $h$ be the vector space and let $\mathbb{H}$ be the symplectic vector space from the construction. For $u\in\mathcal C$, consider the function

$$H_{\hbar}(z,u)=\tilde{H}(z,u)+O(\hbar):\mathfrak{so}_n\longrightarrow \operatorname{End}(\mathbb{H})\llbracket\hbar\rrbracket.$$

Here

$$L^{(2)}GL(h)_\hbar:=\{H_\hbar\in\operatorname{End}(h)((z^{-1}))\llbracket\hbar\rrbracket\mid H_\hbar(-z)^*H_\hbar(z)=1\}$$

is the group of formal symplectic transformations.

Quantum symplecticity conjecture. For any $u\in\mathcal C$, the function $H_{\hbar}(z,u)$ is valued in the group $L^{(2)}GL(h)_\hbar$.

This is a naive conjecture relating the quantum canonical solution of the deformed connection to the symplectic transformations appearing in Givental's formalism. The source indicates that the connection with vertex algebras remains to be explored, and gives no resolution of the conjecture.