The star spanning tree minimum intersection conjecture

Let $G=(V,E)$ be a graph that admits a star spanning tree $T_s$. Let $\mathscr{T}_G$ denote the set of spanning trees of $G$, and let $\cap(T)$ be the intersection number of a spanning tree $T$. Star spanning tree minimum intersection conjecture. For every spanning tree $T\in\mathscr{T}_G$,

$$\cap(T_s)\leq\cap(T).$$

This conjecture generalizes the corresponding result for complete graphs and asserts that a star spanning tree minimizes the intersection number among all spanning trees of any graph admitting one. The surrounding discussion describes reductions that would apply to a hypothetical counterexample, but provides no resolution of the conjecture.