Thin-cylinder variance-order conjecture for first-passage percolation

Let $a_n(h)$ denote the first-passage percolation time across a cylinder of length $n$ and half-height $h$, and let $d\ge 2$. Thin-cylinder variance-order conjecture. The variance satisfies

$$\operatorname{Var}(a_n(h))\asymp nh^{1-d}\qquad\text{for }h=o(n^{1/(d+1)}).$$

The conjecture gives the expected fluctuation scale in the thin-cylinder regime and complements the Gaussian central limit theorem proved there. Its resolution is not stated in the source.