Critical behavior conjecture at the RLD threshold

Consider the random location deaths (RLD) process: species are born with probability $p$, die with probability $q=1-p$, and, on a death event, the least fit species is removed with probability $r$, while otherwise the removal rule uses a uniform random location. For $f\in[0,1]$, let $\mathcal{B}(f)$ be the set of birth times of species with fitness below $f$, and let $B_i$ denote its ordered elements. Define

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

Assume $p>q$. RLD critical-behavior conjecture. If $f=f^\ddagger_c$, then

$$\mathbb{E}[B_{i+1}-B_i]\to\infty\quad\text{as }i\to\infty,$$

and if $f>f^\ddagger_c$, then

$$|\mathcal{B}(f)|<\infty\quad\text{a.s.}$$

This complements the proved subcritical result and predicts the phase transition at $f^\ddagger_c$ for the RLD process. The conjectured assertions remain unproved in the source.