Critical-value conjecture for connectedness of planar self-affine sets

Let $A\in M_2(\mathbb{Z})$ be expanding with $|\det(A)|=3$, let $f$ be its characteristic polynomial, and let ${\mathcal{D}}=\{0,1,b\}v$ with $b>1$ and $v\in\mathbb{R}^2$ such that $\{v,Av\}$ is linearly independent. Write $T(A,{\mathcal{D}})$ for the associated self-affine set. Critical-value conjecture. There exists a critical value $c\geq 2$, dependent on the characteristic polynomial of $A$, such that $T(A,{\mathcal{D}})$ is connected if and only if

$$\frac{c}{c-1}<b\leq c.$$

The theorem preceding this conjecture establishes disconnectedness in several ranges of $b$ for four characteristic-polynomial families, suggesting that the connectedness interval is determined by a single polynomial-dependent threshold; determining the critical values and proving the asserted equivalence remain open.