The fixed-edge extremal conjecture for t-intersecting trees

Let $K_n$ be the complete graph on $n$ vertices, and let a family of spanning trees of $K_n$ be $t$-intersecting if any two trees in the family have at least $t$ edges in common. For a fixed set $E$ of $t$ pairwise disjoint edges, write

$$\mathcal{T}_n(E)=\{T:T\text{ is a spanning tree of }K_n\text{ and }E\subseteq T\}.$$

Fixed-edge extremal conjecture. If $t\le n/2$, then the largest $t$-intersecting family of spanning trees is

$$\mathcal{T}_n(E)$$

for some fixed set $E$ of $t$ disjoint edges.

This is a complete extremal prediction for the range $t\le n/2$. The paper presents it as an open problem, while later results may clarify its relation to the broader conjecture about trivial families.