Unique-solvability conjecture for the input-queued switch functional equation

Consider an input-queued switch operating under a scheduling algorithm that achieves state space collapse. Let $L(\boldsymbol\theta)$ and $\mathbf M(\boldsymbol\theta)$ denote the limiting Laplace transforms appearing in the functional equation for the heavy-traffic distribution of the scaled queue-length vector. Unique-solvability conjecture. The functional equation has a unique solution $(L(\boldsymbol\theta),\mathbf M(\boldsymbol\theta))$ that is a valid Laplace transform of a probability distribution. This uniqueness would establish that the proposed solution is the heavy-traffic distribution of the steady-state scaled queue-length vector; the paper presents it as a conjecture rather than proving it.