Embedding clique-factors in graphs with low ℓ-independence number

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ>0 and integers ℓ,r and n with n∈rN, is it true that there exists an α>0 such that every n-vertex graph G with [Formula presented] and α_ℓ(G)≤αn contains a K_r-factor? We give a negative answer to this question for the case [Formula presented] by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers r,ℓ with [Formula presented] and μ>0, there exist α>0 and N such that for every n∈rN with n>N, every n-vertex graph G with [Formula presented] and α_ℓ(G)≤αn contains a K_r-factor. Here ϱ_ℓ(r−1) is the Ramsey–Turán density for K_r−1 under the ℓ-independence number condition.