Compact linear combinations of composition operators on Hardy spaces

Let ${\phi }_j$, $j=1,2,\dots ,N$, be holomorphic self-maps of the unit disk $\mathbb{D}$ of $\mathbb{C}$. We prove that the compactness of a linear combination of the composition operators $C_{{\phi }_j}:f\mapsto f\circ {\phi }_j$ on the Hardy space $H^p(\mathbb{D})$ does not depend on $p$ for $0<p<\mathrm{\infty }$. This answers a conjecture of Choe et al. about the compact differences $C_{{\phi }_1}-C_{{\phi }_2}$ on $H^p(\mathbb{D})$, $0<p<\mathrm{\infty }$.