The graph-product conjecture for weakly sofic groups

Let $V$ be a set and let $\lbrace \Gamma_v: v \in V \rbrace$ be a family of weakly sofic groups. For a graph $X=(V,E)$, write

$$G_X=\left(\ast_{v\in V}\Gamma_v\right)/\ll h^{-1}k^{-1}hk:(v,w)\in E,\ h\in\Gamma_v,\ k\in\Gamma_w\gg.$$

The graph-product conjecture. The free product $\ast_{v\in V}\Gamma_v$ is weakly sofic, and, more generally, $G_X$ is weakly sofic for every graph $X=(V,E)$ on $V$.

The source says that direct sums of weakly sofic groups are known to be weakly sofic, whereas the corresponding question for other graph products, including free products, appears to be open.