Noetherian type conjecture for Pixley–Roy hyperspaces

Let $(X,\tau)$ be a topological space, and let $PR(X)=[X]^{<\omega}$ be the set of finite subsets of $X$ equipped with the Pixley–Roy topology, whose basic open sets are

$$[F,U]=\{G\in PR(X):F\subset G\subset U\},$$

for $F\in PR(X)$ and $U\in\tau$. Here $Nt$ denotes the Noetherian type of a topological space. Pixley–Roy hyperspace conjecture.

$$Nt(PR(X))\leq Nt(X).$$

This conjecture was tested on several examples of spaces. The analogous inequality $Nt(\operatorname{Exp}(X))\leq Nt(X)$ is known for the Vietoris hyperspace of finite subsets, but the authors identify the Pixley–Roy construction as a possible additional exception to the usual computability of cardinal functions.