Conjecture on semiperfect quasi-$\pi_\infty$-regularity of finitely presented algebra extensions

Let $S\in\mathscr{L}\!\mathscr{T}_2$ be a semiperfect quasi-$\pi_\infty$-regular ring, and let $\varphi:S\to R$ be a ring homomorphism. Regard $R$ as a right $S$-module via $\varphi$, and let $\mathcal{I}_S$ denote the relevant family of ideals. Assume that $R$ is finitely presented and Hausdorff as a right $S$-module, and that for every $r\in R$ and $I\in\mathcal{I}_S$ there is $J\in\mathcal{I}_S$ such that

$$R\varphi(J)r\subseteq R\varphi(I).$$

Semiperfect quasi-$\pi_\infty$-regularity conjecture. Then $R$ is semiperfect and quasi-$\pi_\infty$-regular with respect to some topology. The condition on $J$ ensures that the topology on $R$ generated by cosets of the left ideals $R\varphi(I)$ is a ring topology.