Peskine–Szpiro dimension inequality conjecture

Let $(R,\mathfrak{m})$ be a local ring, and let $\mathfrak{p}$ and $\mathfrak{q}$ be prime ideals in $R$. Assume that $\mathfrak{p}$ has finite projective dimension and

$$\sqrt{\mathfrak{p}+\mathfrak{q}}=\mathfrak{m}.$$

Peskine–Szpiro conjecture. Then

$$\text{dim}\,(R/\mathfrak{p})+\text{dim}\,(R/\mathfrak{q})\leq\text{dim}\, R.$$

The source calls this a famous conjecture concerning finiteness of projective dimensions over regular local rings and cites Peskine and Szpiro. Its resolution status is not supplied in the excerpt.