Totaro's inequality for classifying spaces

For a complex Lie group G and a prime number p, Totaro had conjectured that the dimension of the singular cohomology with Z/p-coefficients of classifying space of G is bounded above by that of the de Rham cohomology of the classifying stack of (the split form of) G in characteristic p. This conjecture was recently proven by Kubrak--Prikhodko. In this note, we give a shorter proof.