The Grassmann-algebraic invariant of 4-dimensional Pachner moves

Let a triangulated $4$-manifold with boundary be given. For every inner edge $ij$, integrate over the Grassmann-algebra expression obtained by multiplying the weights $\tilde{\mathcal W}_{ijklm}$ over all 4-simplices $ijklm$, multiplying by an element $w$, and integrating over the variables $a_{ijkl},b_{ijkl}$ associated with every inner tetrahedron $ijkl$, with denominator $\zeta_{kl}$. Let $d_{ijk}$ be the differential operators associated with the inner 2-faces $ijk$, and define

$$w=\Biggl( \prod_{\substack{\text{over inner}\\ \text{2-faces }ijk}} d_{ijk} \Biggr)^{-1}1.$$

The Grassmann-algebraic Pachner-move invariant. The expression

$$\pm \prod_{\substack{\text{over inner}\\ \text{edges }ij}}\int \prod_{\substack{\text{over all}\\ \text{4-simplices }ijklm}}\tilde{\mathcal W}_{ijklm}\cdot w\cdot\prod_{\substack{\text{over inner}\\ \text{tetrahedra }ijkl}}\frac{\mathrm da_{ijkl}\,\mathrm db_{ijkl}}{\zeta_{kl}}$$

remains invariant under moves $3\to3$ and $2\leftrightarrow4$. This is proposed for systems of deformation parameters satisfying the paper's consistency equations; one stated possibility is to take all such parameters equal to $1$. The claim is preliminary because the choice of interesting consistent systems remains open, and the displayed local identities have only partial verification.