Elementary equivalence from isomorphic cores for graph products

Let $\mathcal G=\mathcal G(\Gamma,\{G_v\}_{v\in\Gamma})$ and $\mathcal G'=\mathcal G(\Delta,\{G'_v\}_{v\in\Delta})$ be graph products with finite graphs, where each vertex group satisfies a simple non-generic almost positive sentence. Core converse conjecture. If there is an isomorphism

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

and $G_v\equiv G'_{f(v)}$, then

$$\mathcal G\equiv\mathcal G'.$$

In particular, each such graph product should be elementarily equivalent to the graph product on its core.