Logarithmically small clique immersions at imm(k)+ε

Determine whether there exists a constant α = α(ε, k) such that every n-vertex graph G with average degree at least imm(k)+ε contains a K_k-immersion on at most α · log n vertices.

Background

Montgomery proved logarithmically small clique minors and subdivisions at the corresponding thresholds minor(k)+ε and sub(k)+ε. An analogous statement for immersions would align the immersion threshold with minor and subdivision results and provide size-sensitive guarantees near the extremal threshold imm(k).