Invariance of torsorial and pseudoétale morphisms under cardinal restriction

Let $S$ be an infinite set, and let $f$ be a $\text{\bf Gen}[S]$-morphism, where $\text{\bf Gen}[S]$ is the subcategory of objects admitting a cover by open sets of cardinality at most that of $S$. The restriction-invariance conjecture.

$$f \text{ is torsorial in }\text{\bf Gen}[S]\Longleftrightarrow f\text{ is torsorial in }\text{\bf Gen},$$

and

$$f \text{ is pseudoétale in }\text{\bf Gen}[S]\Longleftrightarrow f\text{ is pseudoétale in }\text{\bf Gen}.$$

The claim concerns whether restricting the category changes either topology; the source gives no resolution.