Alishahi–Taherkhani's extremal structure conjecture for almost intersecting families

Let $n$, $k$, and $s$ be positive integers with $k\geq 2$, and let $\mathcal{F}\subseteq \binom{[n]}{k}$ be an $s$-almost $1$-intersecting family such that

$$\bigcap_{F\in\mathcal{F}}F=\emptyset.$$

Assume that $n$ is sufficiently large. Alishahi–Taherkhani's conjecture. If $\mathcal{F}$ has maximum size, then

\mathcal{F}\cong\left\\{F\in\binom{[n]}{k}:1\in F,\\ \left|F\cap[k+1]\right|\geq 2\right\\}\cup\mathcal{A}\cup\left\\{[k+1]\setminus\left\\{1\right\\}\right\\},

where $\mathcal{A}$ is an $s$-subset of

\left\\{F\in\binom{[n]}{k}:F\cap[k+1]=\left\\{1\right\\}\right\\}.

This conjectures the extremal structure of maximum-sized almost intersecting families with empty total intersection, extending the asymptotically optimal upper bound established by Alishahi and Taherkhani for $t=1$.