Let $R$ be a local, regular $\mathbb{C}$-algebra and $\mathfrak{m}$ be the maximal ideal. Let $M$ be a finitely generated $R$-module. Suppose there exists $f \in \mathfrak{m}$ such that $M/fM$ is $R/(f)$-torsion-free in the sense that $M/fM$ is torsion-free as a $R/(f)$-module. Is it true that for all but countably many $g \in \mathfrak{m}$ (or $g \in \mathfrak{m}/\mathfrak{m}^2$) we have $M/gM$ is $R/(g)$-torsion-free?