Frankl–Kupavskii stability conjecture for bounded matching number

Let $n,k,t$ be positive integers with $n>kt$. Let $F$ be a $k$-graph of order $n$. For a hypergraph $F$, write $u(F)$ for its matching number, $\tau(F)$ for its vertex-cover number, and $e(F)$ for its number of edges. For $i\in [k]$, let

$${\cal A}_i^{(k)}(n,t)=\left\{A\in {[n]\choose k}: \left|A\cap [(t+1)i-1]\right|\ge i\right\},$$

and let

$${\cal H}^{(k)}(n,t)=\left\{e\in {[n]\choose k}:e\cap [t-1]\ne\emptyset\right\}\cup\left\{[t+k]\setminus [t]\right\}\cup\left\{e\in {[n]\choose k}:e\cap [t]=\\{t\\},\\ e\cap ([t+k]\setminus [t])\ne\emptyset\right\}.$$

Frankl–Kupavskii stability conjecture. If $\nu(F)\le t$, then $\tau(F)\le t$ or

$$e(F)\le\max\\{e({\cal A}_2^{(k)}(n,t)),\dots,e({\cal A}_k^{(k)}(n,t)),e({\cal H}^{(k)}(n,t))\\}.$$

This is a stability strengthening of the Erdős Matching Conjecture, asserting that a $k$-graph with matching number at most $t$ either has a cover of size at most $t$ or is bounded by the extremal non-cover constructions. The supplied text gives no resolution status for this conjecture.