Polynomial edge-connectivity conjecture for tree edge-decompositions

Let $T$ be an arbitrary given tree, and let $G$ be a simple graph. The size of $G$ is $|E(G)|$, and a $T$-edge-decomposition is a decomposition of the edges of $G$ into isomorphic copies of $T$.

Polynomial edge-connectivity conjecture. There are two positive integers $a$ and $n$ such that every $a|E(T)|^n$-edge-connected simple graph whose size is divisible by $|E(T)|$ admits a $T$-edge-decomposition.

This is proposed as a weaker version of the Barát–Gerbner–Thomassé conjecture, motivated by the paper's quantitative theorem. The source does not state a resolution, so the conjecture remains open.