Quantum-disc heat-trace asymptotic conjecture

Let $(\Omega,h)$ be a quantum disc as above, and let $\mathbf{H}(t)$ denote its heat trace, with $c_\gamma$ the leading Weyl-law constant and $\mu(\Omega)$ its quantum area. Quantum-disc heat-trace conjecture. As $t\to0$, one has

$$\mathbb{E}[\mathbf{H}(t)] = \frac{1}{t}c_\gamma\mathbb{E}[\mu(\Omega)] - t^{-1/2+o(1)}.$$

The preceding theorem establishes the analogous exponent formula for domains whose boundary has an inner scaling exponent satisfying the KPZ relation, but the quantum-disc case is proposed as a particularly natural example; the conjecture concerns the expected heat-trace correction and its exponent.