Asymptotic expansion conjecture for TQFT curve operators

Let $M$ be the moduli space, let ${\mathcal T}$ be the parameter space, and let $P$ be a formal trivialization of the formal Hitchin connection, with expansion $P=\sum_{l\geq 0}P^{(l)}h^l$. For any one-dimensional oriented submanifold $\gamma$ and any labeling $\lambda$ of the components of $\gamma$, let $Z^{(n,d)}_k(\gamma,\lambda)$ be the corresponding TQFT curve operator and let $h_{\gamma,\lambda}$ be its associated symbol. Asymptotic expansion conjecture. The mapping class group equivariant formal trivialization of the formal Hitchin connection exists, and

$$Z^{(n,d)}_k(\gamma,\lambda)\sim T^{(k)}_{P(h_{\gamma,\lambda})}.$$

More explicitly, for all $L$ and all $\sigma\in{\mathcal T}$,

$$\left\|Z^{(n,d)}_k(\gamma,\lambda)-\sum_{l=0}^L T^{(k)}_{P^{(l)}_\sigma(h_{\gamma,\lambda})}\frac{1}{(k+n/2)^l}\right\|=O(k^{L+1}).$$

This conjecture predicts that TQFT curve operators admit a full asymptotic expansion governed by the mapping class group equivariant formal trivialization of the formal Hitchin connection. The paper proves existence of the trivialization only to first order, so the all-orders existence and expansion remain open.