Kurano–Roberts symbolic-power intersection conjecture

Let $(R,\mathfrak{m})$ be a regular local ring, and let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals in $R$ such that

$$\sqrt{\mathfrak{p}+\mathfrak{q}}=\mathfrak{m},\qquad \text{dim}\,(R/\mathfrak{p})+\text{dim}\,(R/\mathfrak{q})=\text{dim}\, R.$$

Here $\mathfrak{p}^{(n)}$ denotes the $n$th symbolic power of $\mathfrak{p}$. Kurano–Roberts' conjecture. For all $n>0$,

$$\mathfrak{p}^{(n)}\cap\mathfrak{q}\subseteq\mathfrak{m}^{n+1}.$$

The conjecture extends the result of Kurano and Roberts for regular local rings that contain a field or are ramified. The paper studies it because progress could provide tools for the positivity conjecture for intersection multiplicities; its general case is presented as open.