The Grassmann-algebraic identity for the 2-to-4 Pachner move

Consider the $2\to4$ Pachner move replacing the 4-simplices $(12345)$ and $(12346)$ by $(12356)$, $(12456)$, $(13456)$, and $(23456)$. Let $\tilde{\mathcal W}_{ijklm}$ denote the corresponding Grassmann weights, let $a_{ijkl}$ and $b_{ijkl}$ be the Grassmann variables associated with inner tetrahedra, let $\zeta_{ij}$ be the edge parameters, and let $w_{156,256,356,456}$ be a Grassmann element associated with the four inner 2-faces. The 2-to-4 Pachner-move identity. The following identity holds:

$$\int \tilde {\mathcal W}_{12345} \tilde {\mathcal W}_{12346} \frac{\mathrm da_{1234}\,\mathrm db_{1234}}{\zeta_{34}}$$ $$= - \zeta_{56} \int \tilde {\mathcal W}_{12356} \tilde {\mathcal W}_{12456} \tilde {\mathcal W}_{13456} \tilde {\mathcal W}_{23456} w_{156,256,356,456}$$ $$\times \frac{\mathrm da_{1256} \,\mathrm db_{1256}}{\zeta_{56}} \frac{\mathrm da_{1356} \,\mathrm db_{1356}}{\zeta_{56}} \frac{\mathrm da_{1456} \,\mathrm db_{1456}}{\zeta_{56}} \frac{\mathrm da_{2356} \,\mathrm db_{2356}}{\zeta_{56}} \frac{\mathrm da_{2456} \,\mathrm db_{2456}}{\zeta_{56}} \frac{\mathrm da_{3456} \,\mathrm db_{3456}}{\zeta_{56}}.$$

The identity has been verified for the undeformed part and computationally checked only for selected parameter values in the available verification, so its general validity remains unestablished.