The complete-intersection conjecture for t-intersecting spanning trees

Let $F_{n,\ell}$ be a spanning forest on $n$ vertices with $\ell$ edges, whose components have either $k$ or $k+1$ vertices, with component sizes as equal as possible. Let $\mathcal{F}_{n,t,j}$ be the family of all spanning trees that contain at least $t+j$ of the $t+2j$ edges of $F_{n,t+2j}$. A family of spanning trees is $t$-intersecting if any two trees in it have at least $t$ edges in common.

Complete-intersection conjecture. For any $t$ and $n$, there exists a $j$ such that $\mathcal{F}_{n,t,j}$ is the largest $t$-intersecting set of spanning trees.

This conjecture is presented as a “Complete Intersection Theorem”-type statement, extending the comparison between fixed-edge constructions and families requiring many edges from a balanced spanning forest. The supplied status evidence says that the conjecture has since been proved by Elizaveta Iarovikova and Andrey Kupavskii.