Frankl's quadratic-threshold conjecture for critical intersecting hypergraphs

Let $k>t\ge 1$ be integers, let $\mathcal F$ be a $k$-uniform, $t$-intersecting hypergraph, and let $\tau_t(\mathcal F)$ be the minimum size of a set meeting every edge of $\mathcal F$ in at least $t$ vertices. Write $d=k-t$ and define

$$m(k,t)=\max\{\lvert\mathcal F\rvert:\mathcal F\text{ is }t\text{-intersecting and }t\text{-critical}\},$$

where $t$-critical means $\tau_t(\mathcal F)=k$.

Frankl's conjecture. There exists a constant $c$ such that, whenever $k>c d^2$,

$$m(k,k-d)=\binom{k+d}{d}.$$

The complete $k$-graph on a set of $k+d$ vertices attains the lower bound $\binom{k+d}{d}$. Frankl proved the equality for $k\ge d^4$, and also proved the cases $d=1$ and $d=2$; the conjecture predicts a quadratic threshold in $d$ for the same extremal value.