Establish stability for the modified Erdős–Frankl–Pach conjecture

Prove that, if the modified uniform Erdős–Frankl–Pach conjecture holds and a qualifying family \(\mathcal{F}\) is not a star, then \(|\mathcal{F}|\leq\binom{n-1}{d}-Cn^{d-s}\) for some positive constant \(C\).

Background

The paper proves a strong stability theorem in the case $s=1$, while showing that an analogous statement for $s=d$ with a substantially larger gap cannot hold in that form. It therefore proposes the scale $n^{d-s}$ as the appropriate general stability deficit for the unresolved general conjecture.