Asymptotic crux-based bound for clique subdivisions

Determine whether there exists a constant c > 0 such that for every graph G, if t = min{ d(G), sqrt( c_α(G) / log c_α(G) ) }, where c_α(G) is the α-crux function of G (for a fixed α ∈ (0,1)), then G contains a K_k-subdivision with k ≥ c · t.

Background

The crux c_α(G) measures the minimum order of a subgraph H ⊆ G with average degree at least α·d(G). Im, Kim, Kim, and Liu proved that k = Ω( t * (log log t)^{-6} ), where t = min{ d(G), sqrt( c_α(G) / log c_α(G) ) }, suffices for a K_k-subdivision. The authors note tightness up to polylogarithmic factors and raise whether the (log log t) loss can be removed to achieve k = Ω(t).