Piecewise description conjecture for the radial infinity-harmonic Dirichlet solution

Let $L>1$, and let $R_p$ be the solutions to the $p$-approximation of the Dirichlet problem, with $R=\lim_{p\rightarrow\infty}R_p$. For $R_j=R(s_j)$ and $R'_j=dR/ds(s_j)$, suppose $R$ satisfies the conditions of Proposition 3DIR. Radial solution conjecture. There is a one-parameter family $R_{s^*}$, $s^*\in(0,\infty)$, satisfying those conditions; the Dirichlet data $R(h)=R_0$, with $0<R_0\leq h$, is realized for exactly one $s^*$. The proposed solution is zero on $0\leq s\leq s^*$, then follows the stated linear and ODE pieces. The result is presented as a conjecture because the details were only sketched, and remains open.