Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$?

Are there infinitely many $n$ such that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$?

In [Va99] it is asked whether $968$ is the largest integer with this property, but this is an error, since for example $968-9=7\cdot 137$.

The list of $n$ satisfying the given property is [A214583] in the OEIS. The largest known such $n$ is $1722$.

The answer is negative: [APSSV26b] proves a stronger finiteness theorem, deducing it from Pollack [Po17]. Oriike [Or26] formalised the deduction in Lean.