Gruson's conjecture on Ext of the fraction field of a polynomial ring

Let $k$ be a field. Define $s$ by $|k|=\aleph_s$ if $k$ is infinite, and let $s=0$ otherwise. For a non-negative integer $n$, set $R=k[X_1,\ldots,X_n]$ and let $Q(R)$ denote its fraction field. Gruson's conjecture. One has

$$\operatorname{Ext}_R^i(Q(R),R)\neq 0$$

if and only if $i=\inf\{s+1,n\}$. The paper gives a partial answer to this conjecture by describing minimal pure-injective resolutions in specified cardinality and dimension cases; the general assertion remains open in the supplied context.