Reserved density conjecture for multiple harmonic sums

Let ${\vec{s}}\in{\mathbb N}^l$. Define $\operatorname{density}(RJ({\vec{s}});X)$ to be the proportion, among primes $p$ with $|{\vec{s}}|+2<p<X$, for which $J({\vec{s}}|p)=RJ({\vec{s}})$, and let $\operatorname{density}(RJ({\vec{s}});\infty)$ denote its limiting value.

Reserved density conjecture.

$$\operatorname{density}(RJ({\vec{s}});\infty)= \begin{cases} 1/\sqrt{e}, & \text{if } l=1,\ {\vec{s}}\geq 2,\\ 1/e, & \text{if } l={\vec{s}}=1 \text{ or } l\geq 2. \end{cases}$$

The source attributes this conjecture to Conjecture 7.6 of the author's earlier paper. No proof or resolution is given in the supplied text.