Linear Beer-index bound for p-componentwise simply Δ-connected sets

Call a set $S\subseteq\mathbb{R}^2$ p-componentwise simply $\Delta$-connected if, for every triangle $T$ with $\partial T\subseteq S$, one has $T\subseteq S$. Assume that $\operatorname{b}(S)$ is defined. The linear-bound conjecture. There is an absolute constant $\alpha>0$ such that

$$\operatorname{b}(S)\leq\alpha\operatorname{c}(S).$$

This would extend the linear upper bound proved in the paper for p-componentwise simply connected planar sets to the weaker $\Delta$-connectedness assumption.