Definition: Given $w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x})\in F_m ∗ F_n$ and a group $G$.
The system of equations $\{w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x}) \}$ is solvable in G if $\forall \bar{g}\in G^m$ $\exists \bar{h}\in G^n $ such that $w_i(\bar{g},\bar{h})=1$ for all $ i=1,\cdots,k.$
The system of equations $\{w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x}) \}$ is solvable over G if there exists a group $H$ such that $G\leq H$ for any $\bar{g}\in G^m$ $\exists \bar{h}\in H^n$ such that $w_i(\bar{g},\bar{h})=1$ for all $ i=1,\cdots,k.$ .
My Question: Given a system of equations $\{w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x})\} \in F_m ∗ F_n$ and a group $G_1,G_2$ such that $w_i(\bar{a},\bar{x})$ solvable over $G_1$ and $G_2$ for all $i=1,\cdots, k.$
Is the system of equations $\{w_1(\bar{a},\bar{x}),\cdots, w_k(\bar{a},\bar{x})\}$ solvable over $G_1∗ G_2?$ If it holds, could you give an example of finding a solution in the free product by a non-trivial example? If not, is there any property(from below there is such a set of systems consists of specified equations) (non-trivial) for a given system of equations so that the above question holds, and what is the reason of it holds for the systems consists of equations that is solvable in for all finite alternating groups?
My question arises from there is a characterization of sofic groups $G$ is sofic iff any system of equations solvable in all alternating groups is solvable over G in Lev Glebsky "Approximations of groups, characterizations of sofic groups, and equations over groups", and sofic groups are closed under a free product.
My approaches:
I tried to use an extension of $H_1∗ H_2$ of $G_1∗ G_2$, where $H_i$ is an extension, in which the system is solvable in it, of $G_i.$ But even for the single equation, I couldn't do it.
Also, I tried to construct an extension of $G_1∗ G_2$ such that given an equation $w(a,x)\in F_m ∗ F_n$ and given an element $v(\bar{g_1},\bar{g_2})=\prod_{i=1}^t g_{1_i}g_{2_i}\in G_1∗ G_2$, I could separate the equation $w(v(\bar{g_1},\bar{g_2}),x)=w(\bar{g_1}',x)w(\bar{g_2}',x)$ so that solution commutes ( not exactly of direct product of an extension in which solution exists $H_1,H_2$ of $G_1$ and $G_2$, but idea coming from there.)
Another approach by using another characterization of solvable over, that is given $\bar{a}\in G^m$ $\bar{w}(\bar{a},\bar{x})$ is solvable over $G$ if the following map is a monomorphism
$i:G\rightarrow \frac{G∗ F_n}{ \langle\langle \bar{w}(\bar{a},\bar{x}) \rangle \rangle }$ given by $i(g)=\bar{g}$ for any $g\in G.$
Another way I tried using the free product of the two amalgated free products.