The minimum-degree-three conjecture for strong parity property

A graph is 2-edge-connected if it is connected and remains connected after the deletion of any single edge. A graph has the strong parity property if, for every subset $X\subseteq V(G)$ of even cardinality, it has a parity factor with respect to $X$, namely a spanning subgraph whose vertices in $X$ have odd degree and whose vertices outside $X$ have even degree.

Minimum-degree-three conjecture. Every 2-edge-connected graph of minimum degree at least three has the strong parity property.

This is proposed as a strengthening of the preceding results for graphs with specified cycle structure and for connected $K_{1,r}$-free graphs with minimum degree at least $r\geq 3$. The source presents the statement as an expectation, and no resolution is supplied here.