Asymptotic-velocity conjecture for BBM with uniformly positive rank-based killing

Let $\psi$ be a selection function such that $\psi(x)\geq\delta$ for all $x\in[0,1]$ and $\psi(x)=\delta$ for all $x\in[1-p,1]$ for some $p\in(0,1)$. Asymptotic-velocity conjecture. The $(\psi,1,N)$-BBM has asymptotic velocity

$$v_N^\psi=\sqrt{2(1-\delta)}-\frac{c}{(\log N)^2}+o\left(\frac{1}{(\log N)^2}\right)$$

for some positive constant $c$. This predicts a slower limiting front speed when the rightmost particles are killed at a positive rate; the source gives no proof and does not determine the constant $c$.