Convergence of the derivative at the conjectural quarter-plane-loop radius

Let $Q(a,u)$ be the generating series for quarter-plane loops, let $Q_u(a,u)=\frac{\partial Q}{\partial u}$, and let $\rho_Q(a)$ be the radius of convergence of $Q(a,\cdot)$.

Derivative-convergence conjecture. The series $Q_u(a,u)$ is convergent at $u=\rho_Q(a)$ for $a\geq -1/3$.

This is the third conjecture used in the paper's reduction of the asymptotic problem for deque-sortable and parallel two-stack-sortable permutations. The supplied text does not establish it.