Nonexistence conjecture for star-n idempotent ultrafilters

Let $n\in\mathbb{N}_{>1}$. For ultrafilters on $\mathbb{N}$, define the operation $\star_n$ by extending $a\star_n b=n^a b$ from $\mathbb{N}$ to $\beta\mathbb{N}$. An ultrafilter $p$ is $\star_n$-idempotent if $p\star_n p=p$.

Nonexistence conjecture. For any $n\in\mathbb{N}_{>1}$, $\star_n$-idempotent ultrafilters do not exist.

The conjecture concerns whether the Galvin–Glazer ultrafilter method can be extended to exponential configurations. Such an ultrafilter would yield patterns of the form $\{x,y,xn^y\}$ and could provide an ultrafilter proof of the Sahasrabudhe–Schur theorem; the paper's results only restrict where a hypothetical $\star_n$-idempotent could lie.