The graph-product conjecture for linear sofic groups

Let $\mathbb{K}$ be a field, let $V$ be a set, and let $\lbrace \Gamma_v: v \in V \rbrace$ be a family of $\mathbb{K}$-linear 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 $\mathbb{K}$-linear sofic, and, more generally, $G_X$ is $\mathbb{K}$-linear sofic for every graph $X=(V,E)$ on $V$.

The paper states that closure under direct sums is known, while closure under free products is claimed in the literature with a gap in the proof; the conjecture would establish permanence of $\mathbb{K}$-linear soficity under all graph products.