Weighted Erdős–Burgess lower-bound conjecture for finite rings

Let $R$ be a finite ring with identity, and let $\boldsymbol{I}_{\Psi}(\mathcal{S}_R)$, $\boldsymbol{D}_{\Psi}(\mathop{\rm U}(R))$, ${\rm Ind}(P)$, and ${\rm St}(P)$ denote the weighted Erdős–Burgess constant, weighted Davenport constant, index of a prime ideal, and stabilizer of that ideal under the action of $\Psi$, respectively. Let $\Psi$ be a subgroup of the group ${\rm Aut}(R)$.

Weighted Erdős–Burgess lower-bound conjecture.

$${\rm I}_{\Psi}(\mathcal{S}_R)\geq {\rm D}_{\Psi}({\rm U}(R))+\sum\limits_{P\in \operatorname{spec}(R)} \frac {T\left({\rm Ind}(P);\ \frac{|\Psi|}{|{\rm St}(P)|}\right)} {\frac{|\Psi|}{|{\rm St}(P)|}}.$$

This conjecture proposes a general lower bound for the weighted Erdős–Burgess constant of a finite ring in terms of the weighted Davenport constant of its unit group and a correction term indexed by the prime ideals. The source closes the paper with this conjecture and does not state a resolution.