RLD high-fitness population-density conjecture

Consider the random location deaths (RLD) process with birth probability $p$, death probability $q=1-p$, and parameter $r$. Let $L_n(a)$ be the number of species alive at time $n$ whose fitness is at most $a$, and define

$$f^\ddagger_c=\frac{qr}{p-q(1-r)}.$$

Assume $p>q$. RLD population-density conjecture. For every $f^\ddagger_c\leq a\leq b\leq1$,

$$\frac{L_n(b)-L_n(a)}{n}\stackrel{\mathbb{P}}{\to}(p-q)\left(\frac{b-a}{1-f^\ddagger_c}\right),\qquad\text{as }n\to\infty.$$

This predicts a deterministic limiting density of surviving RLD species above the critical fitness threshold and is presented as an unproved consequence of the conjectured threshold behavior.