Counterexamples without an $R_4$ residue ring

Determine whether there exists a finite ring whose null ideal is not a two-sided ideal and that does not have the ring $R_4$ as a residue ring, where $R_4$ is the order-128 counterexample constructed in the paper.

Background

For every ring $R_n$ constructed in the paper, the authors identify a two-sided ideal $I_n$ such that $R_n/I_n$ is isomorphic to $R_4$. Thus, all members of this counterexample family have the same ring $R_4$ as a residue ring.

The paper leaves unresolved whether finite rings with non-two-sided null ideals exist outside this residue-ring pattern. The question seeks a counterexample not possessing $R_4$ as a residue ring.