Coarse homotopy groups of arbitrary ladders

Let $X$ be a ladder consisting of two rays joined at height $0$ and rungs of lengths $l_m\in\mathbb{N}$ at heights $h_m\in\mathbb{N}$, where $(h_m)$ is strictly increasing, and equip $X$ with its path metric. Let $p_m=2(h_{m+1}-h_m)+l_m+l_{m+1}$ be the perimeter of the $m$-th circle. Ladder conjecture. If $\lim_{m\to\infty}p_m=\infty$, then

$$\pi_n^c(X,\alpha)\cong\begin{cases}0&n\geq1,\\{\varprojlim}^{1}_mF_{\mathbb{N}_{\geq m}}&n=0,\end{cases}$$

where the bonding maps $\psi_{m+1,m}:F_{\mathbb{N}_{\geq m+1}}\to F_{\mathbb{N}_{\geq m}}$ are canonical inclusions. If the perimeters are bounded, then $X$ is coarsely homotopy equivalent to a ray. The statement is known when $l_m\to\infty$; the general case is left open and is expected to depend only on whether the perimeters diverge.