The Kashaev–trisection invariant conjecture for closed 4-manifolds

Let $X$ be an oriented closed $4$-manifold, let $\chi(X)$ be its Euler characteristic, and let $N\geq 2$. Denote by $\mathcal{K}_N(X)$ the Kashaev invariant and by $\tau_{\mathcal{Z}[N]}(X)$ the trisection invariant associated to the Hopf triplet $\mathcal{Z}[N]$. Kashaev–trisection invariant conjecture. For all oriented closed $4$-manifolds $X$ and every $N\geq 2$,

$$\tau_{\mathcal{Z}[N]}(X)=N^{\chi(X)+1}\cdot\mathcal{K}_N(X).$$

The equality holds in the examples computed in the paper, with additional cases checked empirically, but the general statement remains unproved.