Quantum-disc KPZ exponent conjecture for the heat content

Let $(\Omega,h)$ be a quantum disc as defined in the cited work, and let $\mathbf{K}(t)$ denote the corresponding heat-content quantity. Quantum-disc heat-content conjecture. As $t\to0$,

$$\frac{\log \mathbf{K}(t)}{\log t}\to \Delta=\frac{1}{2}$$

in probability. The preceding theorem proves convergence in probability with exponent $\Delta$ under the paper's boundary assumptions, while the quantum-disc case is suggested as a consequence of the same proof ideas and remains conjectural here.