Concavity conjecture for the speed of the maximum of frogs with drift

For $p\in[0,1]$, let the frog model with drift parameter $p$ have maximum speed, namely the limiting speed of its rightmost frog, denoted by $v(p)$. Concavity conjecture. The speed of the maximum is a concave function of $p$; that is, for all $p_1,p_2\in[0,1]$ and $\lambda\in[0,1]$,

$$v(\lambda p_1+(1-\lambda)p_2)\geq \lambda v(p_1)+(1-\lambda)v(p_2).$$

The paper presents this as an open problem motivated by simulations and a heuristic involving a stationary distribution of the number of frogs at the maximum; it does not establish the claimed concavity.