A dimension inequality for Cohen-Macaulay rings

The recent work of Kurano and Roberts on Serre's positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak{n})$ such that $e(A_{\mathfrak{p}})=e(A)$ we have $\dim(A/\mathfrak{p})+\dim(A/\mathfrak{q})\leq\dim(A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak{p}$ is regular.