Quadratic-error edge bound conjecture

Prove that there exists an absolute constant c>0 such that, for every 2-weakly compatible split system S on an n-element set and every associated digraph G_r(S)=(V,E), one has #E≤2#V+cn^2.

Background

The paper’s main graph-theoretic result bounds the number of edges in a digraph representing a 2-weakly compatible split system by (5/2)#V plus a quadratic error term. The conjecture asks whether the leading coefficient can be reduced from 5/2 to 2 while retaining an O(n^2) error term.

Such an improvement would feed into the recurrence argument developed earlier in the paper and yield an O(n^2 log n) upper bound for the maximum size of 2-weakly compatible split systems.