Conjecture on the trigonometric form of curve operators

Let $t=(t_1,\ldots,t_n)$ be a family in $([0,\pi]\cap\mathbb{Q}\pi)^n$, let $D$ be the common denominator of the rational numbers $t_j/\pi$, and let $c_r=(r\frac{t_j}{\pi})\in\mathcal C_r^n$ for $r$ a multiple of $D$. Let $U$ be the parameter domain, let $E$ be the edge set, and let $\gamma$ be a multicurve. A family $T_r\in\operatorname{End}(V_r(\Sigma,c_r))$ is trigonometric if there is an open subset $V\subset U\times[0,1]$ containing $\operatorname{Int}(U)\times\{0\}$ and finitely many smooth functions $F_k\colon V\to\mathbb R$, indexed by maps $k\colon E\to\mathbb Z$, such that $T_r\varphi_{\check c}=\sum_kF_k(\frac{\pi\check c}{r},\frac{\pi}{r})\varphi_{\check c+k}$ for every admissible coloring $\check c$. Trigonometric curve-operator conjecture. For every multicurve $\gamma$, the curve operator $T_r^\gamma$ is trigonometric; its coefficients $F_k$ are recursively computable, $F_k$ vanishes if the geometric intersection of $\gamma$ with some $C_e$ is lower than $k_e$, and $F_k(\mathord\cdot,0)$ is the $k$-th Fourier coefficient of $f_\gamma$ for the $(S^1)^E$-action described in the source. This conjecture predicts a general structural form for TQFT curve operators and relates their leading coefficients to the classical trace function.