Dense minimizer conjecture for induced subgraphs of the Kneser graph

Let $n,k,m\in\mathbb{N}$ satisfy $\binom{n}{k}/2\leq m\leq\binom{n}{k}$, and let $t$ be the smallest natural number such that $\binom{t}{k}\geq m$. For a family $\mathcal{F}\subseteq\binom{[n]}{k}$, write $\Delta(\mathcal{F})$ for the maximum degree of the induced subgraph of the Kneser graph on $\mathcal{F}$.

Dense minimizer conjecture. There exists a family

$$\mathcal{F}\subseteq\binom{[t]}{k}$$

of size $|\mathcal{F}|=m$ such that $\Delta(\mathcal{F})$ is minimum among all subfamilies of $\binom{[n]}{k}$ of size $m$. This is the dense-case analogue of the minimization problem studied in the paper; it is presented as a conjecture and no resolution is given here.