Exact codegree-squared-sum conjecture for t-intersecting families

Let $[n]=\{1,\ldots,n\}$, let $\binom{[n]}{k}$ denote the family of $k$-subsets of $[n]$, and let $\mathcal{F}\subset\binom{[n]}{k}$ be a $t$-intersecting family, meaning that $|F\cap F'|\ge t$ for all $F,F'\in\mathcal{F}$. Write $\operatorname{co}_2(\mathcal{F})$ for its codegree squared sum, and let a $t$-star be the family $\{F\in\binom{[n]}{k}:[t]\subset F\}$.

Exact codegree-squared-sum conjecture. If $t\ge1$ and $n\ge(t+1)(k-t+1)$, then

$$\operatorname{co}_2(\mathcal{F})\le\binom{n-t}{k-t}\bigl(1+(n-k+1)(k-t)\bigr).$$

If $n>(t+1)(k-t+1)$, equality holds only when $\mathcal{F}$ is isomorphic to a $t$-star. This conjecture seeks the codegree-squared-sum analogue of the classical $t$-intersecting Erdős–Ko–Rado theorem; the corresponding bound is proved in the paper for sufficiently large $n$, while the stated optimal range remains open.