Lienardy–Stéphan conjecture for logarithmic bipartite fidelity scalar products

Let $1\leqslant m\leqslant N$ and let $N_1,\dots,N_m\geqslant 1$ be integers satisfying $N=N_1+\dots+N_m$. For the ground-state vectors of the open XXZ chain, define

$$O_{N_1,\dots,N_m}=\langle\psi_N|\left(|\psi_{N_1}\rangle\otimes\cdots\otimes|\psi_{N_m}\rangle\right).$$

The scalar product vanishes if more than one of the integers $N_1,\dots,N_m$ is odd. Scalar-product conjecture. If at most one of the integers $N_1,\dots,N_m$ is odd, then

$$O_{N_1,\dots,N_m}=x^nF_N(x,x^{-1})\prod_{i=1}^m\gamma_{N_i},$$

where $\gamma_{2k}=A_{\mathrm{V}}(2k+1)$ and $\gamma_{2k+1}=N_8(2k+2)$ for each $k\geqslant 0$. This conjectural formula would give closed forms for the logarithmic bipartite fidelity and its multipartite generalisations at finite size; the stated evidence comes from exact computations through $N=12$ sites, and no resolution is supplied here.