The symmetry conjecture for determinants of punctured-hexagon enumeration matrices

Let $a=b$ and $\alpha=\beta$, so the punctured hexagon $H_{a,a,c,\alpha,\alpha,\gamma}$ is symmetric. Let $A$, $C$, and $M$ be the parameters associated with this hexagon, let $N_{a,b,c,\alpha,\beta,\gamma}$ be its enumeration matrix, and let $\mathcal{H}$ denote the hyperfactorial function. If $c$ or $\gamma$ is even, then

$$\det N_{a,b,c,\alpha,\beta,\gamma}=(-1)^{M\left\lceil\frac{C}{2}\right\rceil}\times\frac{\mathcal{H}(M+C)\mathcal{H}(M+\gamma)\mathcal{H}(M+A+\left\lfloor\frac{C}{2}\right\rfloor)\mathcal{H}(M+A+\left\lceil\frac{C}{2}\right\rceil)\mathcal{H}(M+2A+C)}{\mathcal{H}(M+2A)\mathcal{H}^2(M+A+C)\mathcal{H}^2\left(M+\frac{C+\gamma}{2}\right)}$$ $$\times\frac{\mathcal{H}\left(\left\lfloor\frac{M}{2}\right\rfloor\right)\mathcal{H}\left(\left\lfloor\frac{M}{2}\right\rfloor+A\right)\mathcal{H}\left(\left\lfloor\frac{M}{2}\right\rfloor+\frac{C+\gamma}{2}\right)\mathcal{H}\left(\left\lfloor\frac{M}{2}\right\rfloor+A+\frac{C-\gamma}{2}\right)}{\mathcal{H}\left(\left\lfloor\frac{M+C}{2}\right\rfloor\right)\mathcal{H}\left(\left\lfloor\frac{M+\gamma}{2}\right\rfloor\right)\mathcal{H}\left(\left\lfloor\frac{M+C}{2}\right\rfloor+A\right)\mathcal{H}\left(\left\lfloor\frac{M-\gamma}{2}\right\rfloor+A\right)}$$ $$\times\frac{\mathcal{H}\left(\left\lceil\frac{M}{2}\right\rceil\right)\mathcal{H}\left(\left\lceil\frac{M}{2}\right\rceil+A\right)\mathcal{H}\left(\left\lceil\frac{M}{2}\right\rceil+\frac{C+\gamma}{2}\right)\mathcal{H}\left(\left\lceil\frac{M}{2}\right\rceil+A+\frac{C-\gamma}{2}\right)}{\mathcal{H}\left(\left\lceil\frac{M+C}{2}\right\rceil\right)\mathcal{H}\left(\left\lceil\frac{M+\gamma}{2}\right\rceil\right)\mathcal{H}\left(\left\lceil\frac{M+C}{2}\right\rceil+A\right)\mathcal{H}\left(\left\lceil\frac{M-\gamma}{2}\right\rceil+A\right)}$$ $$\times\frac{\mathcal{H}\left(A-\left\lfloor\frac{\gamma}{2}\right\rfloor\right)\mathcal{H}\left(\left\lfloor\frac{C}{2}\right\rfloor\right)\mathcal{H}\left(\left\lfloor\frac{\gamma}{2}\right\rfloor\right)\mathcal{H}\left(A-\left\lceil\frac{\gamma}{2}\right\rceil\right)\mathcal{H}\left(\left\lceil\frac{C}{2}\right\rceil\right)\mathcal{H}\left(\left\lceil\frac{\gamma}{2}\right\rceil\right)}{\mathcal{H}(\gamma)\mathcal{H}^2\left(A+\frac{C-\gamma}{2}\right)}.$$

Moreover, the ideal $I_{a,a,c,\alpha,\alpha,\gamma}=(x^a,y^a,z^c,x^\alpha y^\alpha z^\gamma)$ has the weak Lefschetz property when the characteristic of $K$ is zero or at least $2A+C+M$. This conjecture proposes a closed form for the symmetric case and a corresponding characteristic bound for the weak Lefschetz property.