A threshold conjecture for 3-wise $t$-intersecting uniform families

Let $m(n,k,r,t)$ be the maximum size of an $r$-wise $t$-intersecting family contained in $\binom{[n]}{k}$, where every $r$ members have at least $t$ common elements. The full $t$-star has size

$$\binom{n-t}{k-t}.$$

Threshold conjecture. For

$$n\geq\frac{\sqrt{4t+9}-1}{2}k,$$

one has

$$m(n,k,3,t)=\binom{n-t}{k-t}.$$

This asserts that, in the stated range, the largest 3-wise $t$-intersecting uniform families are full $t$-stars. The supplied text gives no status evidence beyond the conjectural presentation, so it is recorded as open.