Hübl's conjecture on symbolic powers and regular local rings

Let $(R,\mathfrak{m})$ be a regular local ring, let $\mathfrak{p}\in\operatorname{Spec}(R)$ be a prime ideal, let $f\in R$, and let $n$ be an integer such that

$$f^{n-1}\in\mathfrak{p}^{n}.$$

Hübl's conjecture. Then

$$f\in\mathfrak{m}\cdot\mathfrak{p}.$$

The hypothesis implies $f\in\mathfrak{p}^{(2)}$, but the stronger containment in $\mathfrak{m}\mathfrak{p}$ is described as significantly harder; the source does not report a resolution.