Versteegen's even-decomposability conjecture

Let $H$ be a graph with an even number of edges. It is even-decomposable if there is a sequence

$$V(H)=V_0\backslash V_1\backslash\cdots\backslash V_k=\emptyset$$

with each induced graph $H[V_i]$ having an even number of edges and $V_i\backslash V_{i+1}$ an independent set in $H$. Let $r_H(n)$ denote the least number of colours in an edge-colouring of $K_n$ with no even-chromatic copy of $H$.

Versteegen's conjecture. We have

$$r_H(n)=n^{\Omega(1)}$$

if and only if $H$ is even-decomposable.

Versteegen proved the forward obstruction that even-decomposable graphs satisfy $r_H(n)=n^{\Omega(1)}$. Thus the conjecture asserts that this is the only obstruction to the sub-polynomial bound $r_H(n)=n^{o(1)}$; it remains open.