Critical weak Bakry–Émery conjecture for generalized Sierpinski carpets

Let $X$ be a generalized Sierpinski carpet or a similar fractal, with Hausdorff dimension $d_H$, topological-Hausdorff dimension $d_{tH}$, walk dimension $d_W$, and critical exponent $\alpha_1^*$. Let $wBE(\kappa)$ denote the weak Bakry–Émery condition with parameter $\kappa$. The known estimate gives

$$\alpha_1^*\geq \frac{d_H-d_{tH}+1}{d_W}.$$

Generalized Sierpinski carpet criticality conjecture. For generalized Sierpinski carpets and similar fractals,

$$\alpha_1^*=\frac{d_H-d_{tH}+1}{d_W}$$

and $wBE(\kappa)$ holds for some

$$\kappa>(d_W-d_H)_+.$$

The question concerns whether the lower bound for the critical exponent is sharp and whether the weak Bakry–Émery estimate can be improved beyond the currently known parameter. The associated open problem asks under which conditions $wBE(d_W-d_H+d_{tH}-1)$ holds and the critical assumption is satisfied.