Polynomial degree bound for Betti numbers of graded quotients

Let $S$ be a standard graded polynomial ring over the field $k$, and let $I\subset S$ be a homogeneous ideal generated in degree $d$ with projective dimension $p$. The Betti numbers $\beta_i(S/I)$ are the ranks occurring in the minimal graded free resolution of $S/I$ over $S$.

Polynomial Betti-bound conjecture. All Betti numbers of $S/I$ are bounded, as functions of $d$, by polynomials of degree at most $p-1$.

The preceding results establish polynomial bounds in several cases and motivate this general assertion. Its status is not determined by the supplied text.