Asymptotic equality frequencies for Hilbert depths of path and cycle ideals

Let $S$, $I_n$, and $J_n$ be as above, and for each $N$ let $\#\{n\leq N:\,P(n)\}$ denote the number of positive integers $n\leq N$ satisfying $P(n)$. Hilbert-depth frequency conjecture.

$$\lim_{N\to\infty}\frac{\#\{n\leq N:\,\operatorname{hdepth}(S/I_n)=\operatorname{hdepth}(S/J_n)\}}{N}=\frac23,$$

and

$$\lim_{N\to\infty}\frac{\#\{n\leq N:\,\operatorname{hdepth}(I_n)=\operatorname{hdepth}(J_n)\}}{N}=\frac56.$$

The source presents these limits as further proposals after reporting verification of the preceding conjecture for $n\leq 1000$; no proof or disproof of either asymptotic assertion is reported.