The subprincipal-symbol conjecture for curve operators

Let $\gamma$ be a multicurve, let $f_\gamma$ be its trace function, and let $(\tau,\theta)$ be action-angle coordinates on the relevant open subset of $\mathcal M(\Sigma,t)$, with $\tau$ in the interior of $U$ and $\theta\in(\mathbb R/2\pi\mathbb Z)^E$. For the $\psi$-symbol $\sigma^\gamma=\sum_kF_k(\tau,\hbar)e^{ik\theta}$, where $k\theta=\sum_ek_e\theta_e$, Subprincipal-symbol conjecture. The symbol has the asymptotic development

$$\sigma^\gamma=f_\gamma+\left(\frac{1}{2i}\sum_e\frac{\partial^2f_\gamma}{\partial\theta_e\partial\tau_e}\right)\hbar+o(\hbar).$$

This is the proposed first-order deformation formula for the curve-operator symbol, identifying its subprincipal term with mixed action-angle derivatives of the classical trace function.