The upper bound is $‖T‖_c ≤ C_k ρ_k(T)$ for each $k$. For the lower bound, Kingman's theorem is applied to covering radii of the image of the unit ball, which are measurable and subadditive and lie below $‖·‖_c$; the compactness seminorm itself is not known to be measurable in $ω$. The counting argument of Lemma 34 of Lee (2024) then compares the two.