The extremal-family conjecture for shades of t-intersecting families

For $1\le t\le k\le m\le n$, let $I(n,k,t)$ be the family of all $t$-intersecting subfamilies of $\binom{[n]}{k}$, and let

$$M_0(n,m,k,t)=\max_{\mathcal A\in I(n,k,t)}|\nabla_m(\mathcal A)|,$$

where $\nabla_m(\mathcal A)$ is the $m$-shade of $\mathcal A$. Define

$$\mathcal F_i(n,m,t)=\left\{F\in\binom{[n]}{m}:|F\cap[t+2i]|\ge t+i\right\}.$$

The shade extremal-family conjecture. One has

$$M_0(n,m,k,t)=\max_{0\le i\le\min\left(k-t,\frac{n-t}{2}\right)}|\mathcal F_i(n,m,t)|.$$

The conjecture proposes that the largest shade is attained by one of the standard extremal families. The paper notes that the analogous assertion is false, since the optimal family for the $4m$-conjecture yields an asymptotic ratio of $1/2$ in a case where the proposed bound would not allow it.