Non-concavity conjecture for the critical curve of long-range percolation

Let $d\geq 2$, let $k>1$, and consider the critical curve $p\mapsto q_c(p)$ for the case $m=2$ and $l=1$ of long-range percolation on an oriented $d$-ary tree. For a function $q_c$, concavity on an interval means that its graph lies above every chord joining two points of the graph.

Non-concavity conjecture. There exists $\delta=\delta(k,d)>0$ such that $p\mapsto q_c(p)$ is not concave on $[0,\delta]$.

This claim formalizes the observation that the critical curve, contrary to an intuitive concave depiction, appears convex near $p=0$ when $l=1$ and $k\geq 2$. The supplied text does not state whether the claim has been proved or remains open.