Zhang's finiteness conjecture for the critical passage-time exponent

Let $F_a$ be the distribution defined by

$$F_a(x)= \begin{cases} 1&\text{if }x^a>1-p_c,\\ x^a+p_c&\text{if }0\leqslant x^a\leqslant 1-p_c,\\ 0&\text{if }x<0, \end{cases}$$

where $p_c$ is the critical probability for Bernoulli bond percolation on $\mathbb{Z}^2$, and let $\rho(F_a)=\lim_{n\to\infty}T(\mathbf{0},\partial B(n))$, with $B(n)=\{x\in\mathbb{Z}^2:\|x\|_\infty\leqslant n\}$ and $\partial B(n)=\{x\in\mathbb{Z}^2:\|x\|_\infty=n\}$. Zhang's conjecture. The quantity

$$\sup\{a>0:\rho(F_a)<\infty\}$$

is finite. Zhang proved that this quantity is nonempty because $\rho(F_a)<\infty$ almost surely for sufficiently small $a$; the conjecture asserts that finiteness fails for all sufficiently large $a$.