The generalized right-tight conjecture on essential socles

Let $R$ be a ring, let $E(R_R)$ be the injective envelope of the right regular module, and call $R$ right generalized $R$-tight when every cyclic submodule of $E(R_R)$ embeds in a free module. A ring is right Kasch when every simple right $R$-module embeds in $R$. Generalized right-tight socle conjecture. If $R$ is a right Kasch right generalized $R$-tight ring, then $R_R$ has finitely generated essential socle. This extends the CF and FGF conjectures and is presented as a new open problem; the source gives partial positive results under additional assumptions.