The polynomial tail lower-bound conjecture for spiral-like domains

Let $W\neq\mathbb C$ be a simply connected spiral-like planar domain containing $0$, let $T_W$ denote the first exit time of planar Brownian motion from $W$, and define

$${\rm H}(W)=\sup\{p>0:{\bf E}[(T^W)^p]<\infty\}.$$

Polynomial tail conjecture. For every $p>{\rm H}(W)$, there exists a constant $C>0$ such that

$${\bf P}(T_W>t)\geq\frac{C}{t^p}.$$

The authors state that this bound would imply the preceding long-stay limit conjecture for spiral-like domains. It remains unproved in the paper.