  Let $\varphi_j$, $j=1,2, \dots, N$, 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 $C_{\varphi_j}: f\mapsto
f\circ\varphi_j$ on the Hardy space $H^p(\mathbb{D})$ does not depend on $p$
for $0<p<\infty$. This answers a conjecture of Choe et al. about the compact
differences $C_{\varphi_1} - C_{\varphi_2}$ on $H^p(\mathbb{D})$, $0<p<\infty$.
