The sharp formula for non-trivially $t$-intersecting families

For integers $r\ge 3$ and $r-2\ge t\ge 1$, let $\iota_0(t,n;r)$ denote the maximum size of a $t$-intersecting family in an $r$-partite $r$-uniform hypergraph with all parts of size $n$ and with total intersection of size less than $t$. The sharp-formula conjecture. If $n\ge 2$, then

$$\iota_0(t,n;r)=\max\\{n^{r-t}-(n-1)^{r-t}+t(n-1),(t+2)n^{r-t-1}-(t+1)n^{r-t-2}\\}.$$

For sufficiently large $n$, the paper proves this formula and identifies a phase transition at $t=\frac{r}{2}-1$; the conjecture asserts that the same extremal formula holds for every $n\ge 2$.