Choe–Choi–Kim–Park independence-of-p conjecture for differences of composition operators

Let $\mathbb D$ be the unit disk, let $\mathcal{S}(\mathbb D)$ denote the holomorphic self-maps of $\mathbb D$, and for $\varphi\in\mathcal{S}(\mathbb D)$ let $C_\varphi f=f\circ\varphi$. For $0<p<\infty$, let $H^p(\mathbb D)$ be the Hardy space on $\mathbb D$. Choe–Choi–Kim–Park's conjecture. For any $\varphi,\psi\in\mathcal{S}(\mathbb D)$, the compactness of

$$C_\varphi-C_\psi:H^p(\mathbb D)\to H^p(\mathbb D)$$

is independent of the parameter $p\in(0,\infty)$. The conjecture extends the known independence for $1\leq p<\infty$ to the full range $0<p<\infty$.