Briançon–Skoda number polynomial-extension conjecture

Let ${\bf R}$ be a Noetherian ring of finite Krull dimension. Define the Briançon–Skoda number of ${\bf R}$ by

$$c({\bf R})=\text{\rm sup}\{c(I)\mid I\subset {\bf R}\}.$$

Polynomial-extension conjecture. The Briançon–Skoda number satisfies

$$c({\bf R}[x])\leq c({\bf R})+1.$$

The conjecture asks how the normalization behavior measured by the Briançon–Skoda number changes under adjoining one polynomial variable; no resolution is given in the source.