Conjectured radius of convergence for the quarter-plane loop series

Let $Q(a,u)$ be the generating series for quarter-plane loops, and write $Q(a,\mathord\cdot)$ for the resulting univariate series in $u$. For $a\ge -1$, let $\rho_Q(a)$ denote its radius of convergence. Conjectured radius formula.

$$\rho_Q(a)=\begin{cases}\displaystyle \frac{1}{(2+\sqrt{2+2a})^2},&a\ge -\frac12,\\[6pt]\displaystyle -\frac{a}{2(a-1)^2},&a\in[-1,-\frac12].\end{cases}$$

The formula was supported numerically and by analytic predictions, but the source states it as a conjecture; only the range $a\ge -1/3$ is needed in the subsequent analysis.