Converse Lyapunov conjecture for robust delay stability

Fix a network with queue $m$, tail-exponent set $\Gamma$, and the notions of RDS and special $\epsilon$-Lyapunov function as in the preceding corollary. Converse Lyapunov conjecture. If queue $m$ is RDS for every tail exponent ${\boldsymbol{\gamma}}\in\Gamma$, then there exists some $\epsilon>0$ and a special $\epsilon$-Lyapunov function. This would strengthen part (b) of the corollary to a converse of part (a); whether the converse holds is left open.