Line-middle graph inequality conjecture for outer-connected domination

Let $G$ be a graph of order $n\ge 2$, and let $L(G)$ and $M(G)$ denote its line graph and middle graph, respectively. The line-middle graph inequality conjecture asserts that

$$\gamma{\tilde{}} _{c}(L(G)) \le \gamma{\tilde{}} _{c}(M(G)).$$

The inequality is supported by the preceding results, including strict inequality for wheels of order at least five; equality occurs for at least one connected graph of order four. A classification of graphs for which equality holds is posed separately as an open problem, while the general inequality remains open.