Strong stability of 3-wise $t$-intersecting families

Let $\mathcal{G}$ be a family of subsets of an $n$-element set. The family $\mathcal{G}$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal{G}$ is of size at least $t$. For a real number $p\in (0,1)$ we define the measure of the family by the sum of $p^{\mathrm{\mid }G\mathrm{\mid }}(1-p{)}^{n-\mathrm{\mid }G\mathrm{\mid }}$ over all $G\in \mathcal{G}$. For example, if $\mathcal{G}$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. Let $0<p\le 2\mathrm{/}(\sqrt{4t+9}-1)$, $\delta >0$, and let $\mathcal{G}$ be a $3$-wise $t$-intersecting family. It is known that the measure of $\mathcal{G}$ is at most $p^t$. Suppose, moreover, that $\mathcal{G}$ has the measure at least $(\frac{1}{2}+\delta )p^t$. We show that, by choosing $t$ sufficiently large depending on $\delta$, the structure of $\mathcal{G}$ is one of (i) and (ii): (i) every subset in $\mathcal{G}$ contains a fixed $t$-element set, (ii) every subset in $\mathcal{G}$ contains at least $t+2$ elements from a fixed $(t+3)$-element set.