Mock Jacobi property of the generating function for planar hexagons

Let $f_5(\boldsymbol{z};\tau)$ be the generating function defined by

$$f_5(\boldsymbol{z}; \tau):=\sum_{\boldsymbol{n}\in \mathbb{Z}^4 \cap D }\operatorname{sgn}\left(n_1+\alpha_{1}-n_3-\alpha_{3}\right) q^{ 3 n_1n_2-\frac{3}{2}n_3^2-\frac{3}{2}n_4^2} \zeta_1^{3n_2} \zeta_2^{3n_1} \zeta_3^{-3n_3} \zeta_4^{-3n_4},$$

where the parameters $\alpha_k$, for $k\in\{1,2,3,4\}$, satisfy $\lvert \alpha_3\rvert,\lvert \alpha_4\rvert\leq\min(\lvert \alpha_1\rvert,\lvert \alpha_2\rvert)$, and

$$D:=-\boldsymbol{\alpha}+\left\{\boldsymbol{x}\in\mathbb{R}^4: \lvert x_3\rvert,\lvert x_4\rvert\leq\min(\lvert x_1\rvert,\lvert x_2\rvert)\text{ and }x_1x_2,x_1x_3,x_1x_4\geq 0\right\}.$$

Here $q$ and $\zeta_j$ are the variables used in the generating function. Mock Jacobi property conjecture. The generating function $f_5$ has mock Jacobi properties. This conjecture extends the mock modular behavior established for the generating functions of polygons with fewer sides and is motivated by homological mirror symmetry; the precise completion and transformation laws remain to be determined.