Three-point higher-order conformal divergence conjecture

Fix $i\in\{1,2,3\}$. Let $\mathbf{s}\in\mathbb{Z}_{\geq 0}^3$, $s_i>0$, $\Delta_j\in\mathbb{C}$ for $j\neq i$, and let the depth satisfy $t_i\in\{0,\ldots,s_i-1\}$. For non-negative exponents $h_1,h_2,h_3,v_1,v_2,v_3$, consider the three-point tensor structure built from $H_{12},H_{23},H_{13}$ and $V_{1,23},V_{2,31},V_{3,12}$, with the usual powers of $P_{12},P_{23},P_{13}$ determined by the dimensions. Three-point higher-order conformal divergence conjecture. There exists a differential operator $\mathcal{D}_{i,s_i-t_i}$ acting on $\mathbb{C}[H_{ij},V_{i,jk}]$ such that

$$(\partial_{P_i}\cdot D_{Z_i})^{s_i-t_i}\left(f\left(\frac{H^{h_1}_{12}H^{h_2}_{23}H^{h_3}_{13}V^{v_1}_{1,23}V^{v_2}_{2,31}V^{v_3}_{3,12}}{P^{\tau_{12,3}}_{12}P^{\tau_{23,1}}_{23}P^{\tau_{31,2}}_{13}}\right)\right)=f\left(\frac{\mathcal{D}_{i,s_i-t_i}(H^{h_1}_{12}H^{h_2}_{23}H^{h_3}_{13}V^{v_1}_{1,23}V^{v_2}_{2,31}V^{v_3}_{3,12})}{P^{\tau_{12,3}}_{12}P^{\tau_{23,1}}_{23}P^{\tau_{31,2}}_{13}}\right),$$

if and only if $\Delta_i=d-1+t_i$, where

$$v_1+h_1+h_3=s_1,\qquad v_2+h_1+h_2=s_2,\qquad v_3+h_2+h_3=s_3.$$

This is the proposed three-point extension of the preceding first-order differential-operator theorem.