Degree-power extremal conjecture for bounded-matching families

Let $k\geq2$, $s\geq1$, $1\leq r\leq k-1$, and let $p\geq1$ be real. Suppose that

$$n\geq(2s+1)k-s.$$

If $\mathcal F\subseteq\binom{[n]}k$ satisfies $\nu(\mathcal F)\leq s$, let $\mathcal B_S$ denote the family of all $k$-sets meeting a fixed $s$-set $S$, and let $\ell_{r,p}$ be the degree-power quantity used in the paper. Bounded-matching degree-power conjecture.

$$\ell_{r,p}(\mathcal F)\leq\ell_{r,p}(\mathcal B_S).$$

Moreover, equality holds if and only if $\mathcal F$ is isomorphic to $\mathcal B_S$. This extends the paper's bounded-matching result from codegrees to arbitrary nontrivial degree levels.