Symbolic-power intersection conjecture SP-2

Let $(R,\mathfrak{m})$ be a regular local ring with prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ such that $\sqrt{\mathfrak{p}+\mathfrak{q}}=\mathfrak{m}$ and

$$\operatorname{dim}(R/\mathfrak{p})+\operatorname{dim}(R/\mathfrak{q})=\operatorname{dim}(R).$$

SP-2 conjecture. The intersection of symbolic powers satisfies

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

for all $m,n\geq 1$. The source presents this as a conjectural generalization of the preceding symbolic-power statement; the supplied context does not establish its resolution.