Compact linear combinations of composition operators on Hardy spaces

Let φⱼ, j=1,2, , N, be holomorphic self-maps of the unit disk D of C. We prove that the compactness of a linear combination of the composition operators Cφⱼ: f↦ f∘φⱼ on the Hardy space Hᵖ(D) does not depend on p for 0<p<∞. This answers a conjecture of Choe et al. about the compact differences Cφ₁ - Cφ₂ on Hᵖ(D), 0<p<∞.