Cycle-spectrum conjecture for the associahedron

Let $\cG_N$ be the associahedron graph, let $S(\cG_N)$ be the set of lengths of cycles in $\cG_N$, and let $\cD_{N,3}$ denote the comparison set used in the source. Cycle-spectrum conjecture. Almost all lengths are possible, more precisely

$$\frac{|S(\cG_N)|}{|\cD_{N,3}|}=1-o(1)\quad\text{as }N\to\infty.$$

The conjecture asks for the asymptotic size of the cycle spectrum of the associahedron; the supplied source gives no resolution status.