Elementary classification by cores for graph products in the specified classes

Let $\mathcal G(\Gamma,\{G_v\}_{v\in\Gamma})$ and $\mathcal G(\Gamma',\{G'_v\}_{v\in\Gamma'})$ belong to the specified class $\mathcal G(\mathbf F_N,\mathbf C^r_\Phi)$ with $N>0$. Assume either that $\Phi$ consists only of nontrivial positive sentences and the graphs are positive reduced, or that $\Phi$ consists of non-generic almost positive sentences and the graphs are almost positive reduced. Graph-product core classification conjecture. Then

$$\mathcal G(\Gamma,\{G_v\}_{v\in\Gamma})\equiv\mathcal G(\Gamma',\{G'_v\}_{v\in\Gamma'})$$

if and only if there is an isomorphism

$$f:\operatorname{Core}(\Gamma)\to\operatorname{Core}(\Gamma')$$

and $G_v\equiv G'_{f(v)}$. This would extend the RAAG core principle to the indicated classes of graph products.