Eventual monotonicity of column-number functions

Determine whether the counting functions \(\g(\Delta,r)\), \(\s(\Delta,r)\), and \(\h(\Delta,r)\) are eventually non-decreasing when either the number of rows \(r\geq3\) is fixed or the modularity parameter \(\Delta\) is fixed.

Background

The paper proves that $\g(\Delta,2)$ is eventually non-decreasing and notes that existing results give affirmative answers in several other fixed-parameter cases. However, the authors ask whether eventual monotonicity holds generally for the generic, simple, and unrestricted column-number functions in higher dimensions or at fixed modularity parameter.