Linear sofic free-product profile conjecture

Let $G$ and $H$ be linear sofic groups, let $G\ast H$ be their free product, and let $F_2$ denote the free group of rank $2$. Write $\mathcal{D}^{lin}_G(n)$ for the linear sofic profile and $\Phi_{F_2}(n)$ for the residual finiteness growth of $F_2$, with $\preccurlyeq$ denoting the profile-comparison relation. The linear sofic free-product conjecture.

$$\mathcal{D}^{lin}_{G\ast H}(n)\preccurlyeq \left(\mathcal{D}^{lin}_{G}(n)+\mathcal{D}^{lin}_{H}(n)\right)\Phi_{F_2}(n).$$

The source introduces this as an unexplored extension of its free-product estimates: it gives an upper bound for sofic profiles, while the corresponding linear sofic bound is asserted here without supplied evidence of resolution.