Goddyn's thin tree conjecture

Let $G=(V,E)$ be an unweighted undirected graph. A set $T\subseteq E$ is $\alpha$-thin with respect to $G$ if, for every nonempty set $S\subsetneq V$,

$$\lvert T(S,\overline{S})\rvert\leq\alpha\lvert E(S,\overline{S})\rvert.$$

A graph is $d$-edge-connected if every cut has at least $d$ edges. Thin Tree Conjecture. For any $\alpha<1$, there exists $d\geq1$ such that any $d$-edge-connected graph $G$ has a spanning tree $T$ that is $\alpha$-thin. This conjecture asks for combinatorial constructions of thin spanning trees; spectral constructions of linear-sized thin subsets are known, but the combinatorial question remains open.