Critical cylinder exponent conjecture for first-passage percolation

Let $a_n(h)$ be the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and define

$$\gamma_c(d):=\sup\left\{\gamma:\frac{a_n(n^\gamma)-\operatorname{\mathds{E}}[a_n(n^\gamma)]}{\sqrt{\operatorname{Var}(a_n(n^\gamma))}}\Longrightarrow N(0,1)\text{ as }n\to\infty\right\}.$$

For distributions satisfying the stated moment and admissibility conditions, one has $\gamma_c(d)\ge 1/(d+1)$. Critical cylinder exponent conjecture. For any $d\ge 2$,

$$\gamma_c(d)=\frac{1}{d+1}.$$

The conjecture asserts that the currently known lower bound is sharp; the source notes that there are indications for this, particularly in dimension two.