Critical percolation distributional complexity exponent conjecture

Let $G_m$ be the square-lattice percolation graph in an $m$-by-$m$ square, let $f_m$ be the Boolean function indicating an open left-to-right crossing, and let $\pi_{1/2}$ denote critical percolation. The quantities $s(f_m,\pi_{1/2})$, $w(f_m,\pi_{1/2})$, $\ell(f_m,\pi_{1/2})$, and $a(f_m,\pi_{1/2})$ are the distributional sensitivity, witness, randomized decision-tree, and algorithmic complexities, respectively.

Critical percolation complexity conjecture. As $m\to\infty$,

$$s(f_m,\pi_{1/2})=m^{3/4+o(1)},$$ $$w(f_m,\pi_{1/2})\le m^{4/3+o(1)},$$ $$\ell(f_m,\pi_{1/2})\ge m^{3/2+o(1)},$$

and

$$a(f_m,\pi_{1/2})\le m^{7/4+o(1)}.$$

These predictions arise from the conjectured four-, three-, and two-arm exponents $5/4$, $2/3$, and $1/4$, respectively; determining the exact exponents in the relevant cases is a well-known open problem, while the corresponding conformal-invariance results are known for critical site percolation on the triangular lattice.