The connected-graph bound for eternal connected domination

Let $G$ be a connected graph on $n$ vertices, let $\Delta(G)$ denote its maximum degree, let $\theta(G)$ denote its clique covering number, and let $\gamma_c^{\infty}(G)$ denote its eternal connected domination number. The eternal connected domination conjecture. For every connected graph $G$ with

$$\Delta(G)<n-1,$$

one has

$$\gamma_c^{\infty}(G)>\theta(G).$$

The source lists this among open problems in the other eternal-protection models.