Intersecting families of loose spanning hypertrees

A loose hypertree is a hypergraph whose incidence graph is a tree. Let $b\ge3$, $n\equiv1\pmod{b-1}$, and $s:=(n-1)/(b-1)$. We determine the largest $t$-intersecting families of loose spanning $b$-uniform hypertrees on $[n]$, simultaneously for every $1\le t\le s$, provided $n$ exceeds an explicit c