For every nonnegative integer k, there is a cograph on at least $3^k$ and at most $4^k$ vertices such that every spanning tree has crossing number at least $ceil(k/2)$ for the open-neighborhood set system.

The witnesses are the same recursively constructed cographs that require k crossings in every Welzl order. Every spanning tree of crossing number c can be linearized with crossing number at most $2c$: delete leaves one at a time and, in reverse, insert each leaf beside its tree neighbor. Inserting a leaf changes a row only when its incident tree edge is crossed, and then adds at most two crossings. Hence an order lower bound of k gives a tree lower bound of $ceil(k/2)$.

Crespelle and Gambette proved the corresponding asymptotic lower bound in the equivalent language of cograph contiguity, using cographs with complete binary cotrees. The theorem below records the consequence for arbitrary Welzl spanning trees with explicit finite-size constants inherited from the preceding formalized order obstruction.