Extension to shifted-prime progressions for polynomials with nonzero constant term

Determine whether the quantitative results known for polynomial progressions involving polynomials with nonzero constant term can be extended to progressions whose difference parameter is restricted to the shifted primes b\mathbb P-1d.

Background

The paper establishes quantitative polynomial Szemer di-type bounds for shifted-prime differences in settings modeled on the BergelsonLeibman theorem, with the polynomials required to have zero constant term. The authors contrast this scope with results of PeluseSahSawhney and KravitzKucaLeng for progressions involving polynomials such as $y^2-1$ and, more generally, polynomials having a simple integer root.

The unresolved issue is whether analogous quantitative existence results can be proved when the polynomial difference is evaluated at $p-1$, with $p$ prime, for polynomial families outside the zero-constant-term setting. This is explicitly left open rather than resolved by the paper.