Ellis–Keller–Lifshitz stability conjecture for intersecting families

Let $n$, $k$, $t$, and $d$ satisfy

$$n\geq(t+1)(k-t+1),\qquad d\in\mathbb{N}.$$

Let $\mathcal{F}\subseteq\binom{[n]}{k}$ be $t$-intersecting. Ellis–Keller–Lifshitz stability conjecture. If

$$|\mathcal{F}|\geq\max\left\{(t+2)\binom{n-t-2}{k-t-1}-(t+1)\binom{n-t-2}{k-t-2}, \binom{n-t}{k-t}-\binom{n-t-d}{k-t}+t\binom{n-t-d}{k-t-d+1}\right\},$$

then there exists a $t$-subset $X$ such that

$$|\mathcal{F}\setminus\mathcal{F}[X]|\leq t\binom{n-t-d}{k-t-d+1}.$$

Here $\mathcal{F}[X]$ denotes the subfamily of members of $\mathcal{F}$ containing $X$. The conjecture was proposed by Ellis, Keller and Lifshitz as a strengthening of their stability theorem for $t$-intersecting families. The supplied text does not state whether it has been resolved.