The mixed edge-deletion conjecture for majority C-chromatic edge-critical graphs

Let $G$ be an $\overline{\chi}_{\geqslant}$-edge-critical graph with at least two edges, and let $e,f\in E(G)$. The mixed edge-deletion conjecture. It is impossible that

$$\overline{\chi}_{\geqslant}(G-e)<\overline{\chi}_{\geqslant}(G)<\overline{\chi}_{\geqslant}(G-f).$$

The claim concerns whether edge deletions from an edge-critical graph can produce both a decrease and an increase in the majority $C$-chromatic number. The supplied text does not state whether this has been resolved.