Removal of torsion hypotheses in the asymptotic regularity theorem for subquotients

Determine whether the conclusion of Theorem 4.1 for the regularity of \((U+\mathbf{I}^{\underline n}V)/\mathbf{I}^{\underline n}W\) remains valid without assuming that \(U\) is \(R_+\)-torsion or that \(\mathbf{I}^{\underline c}V/\mathbf{I}^{\underline c}W\) is \(R_+\)-torsion for some \(\underline c\in\mathbb N^r\).

Background

Theorem 4.1 establishes a dichotomy for the eventual regularity of $$(U+\mathbf{I}^{\underline n}V)/\mathbf{I}^{\underline n}W$$: under one of two $R_+$-torsion hypotheses, the modules are eventually either zero or nonzero, and in the nonzero case their regularity is either eventually $\reg(U)$ or the supremum of finitely many linear functions of $\underline n$.

The paper leaves unresolved whether the same dichotomy and asymptotic description hold when both torsion assumptions are removed.