Connectivity threshold conjecture for honest connected colorings

Let $\mathcal{P}$ be an honest property of sequences, meaning that it is inherited by every nonempty block, is preserved under concatenation of sequences over disjoint color sets, and admits arbitrarily long sequences over a finite alphabet. Let $m(\mathcal{P})$ be the least alphabet size for which arbitrarily long sequences with property $\mathcal{P}$ exist, and let $\operatorname{\mathcal{P}-c}(G)$ be the minimum number of colors in a $\mathcal{P}$-connected coloring of $G$.

Connectivity threshold conjecture for honest connected colorings. For every honest property $\mathcal{P}$, there exists a constant $c(\mathcal{P})$ such that every $c(\mathcal{P})$-connected graph $G$ satisfies

$$\operatorname{\mathcal{P}-c}(G)\leq m(\mathcal{P}).$$

The claim asks whether sufficiently high connectivity always permits the optimal sequence alphabet size. The paper gives a related bound for $4$-connected graphs, but leaves this stronger formulation open.