Use the cograph from the logarithmic Welzl-order lower bound. Given any spanning tree of crossing number $c$, repeatedly delete a leaf and reinsert it beside its unique neighbor to obtain an order of crossing number at most $2c$. The witness requires at least $k$ order crossings, so $k ≤ 2c$, which is equivalent over the naturals to $ceil(k/2) ≤ c$.

Crespelle and Gambette proved the logarithmic contiguity lower bound for cographs with complete binary cotrees. The explicit order witnesses used here are imported from lax-214022. The leaf-removal linearization is the standard depth-first factor-two conversion from tree cuts to a linear layout, and upgrades their order phenomenon to arbitrary spanning trees.