Determine whether combinatorics distinguish free and nearly free arrangements in the exceptional range

Determine whether there exist a free line arrangement A and a nearly free line arrangement B with the same combinatorics, specifically in the case d1 = m + 1 and 2m + 3 ≤ d, where m denotes the maximal multiplicity and d the number of lines.

Background

The paper discusses the extent to which the freeness of a line arrangement can be decided from its combinatorics. It identifies d1 = m + 1 with 2m + 3 ≤ d as a regime in which freeness is not readily determined by the available numerical criteria.

The authors exhibit two arrangements with identical global Tjurina number: a free arrangement with exponents (m + 1, m + 1) and a nearly free arrangement with exponents (m, m + 3, m + 3). However, they do not establish whether such arrangements can have the same combinatorics, leaving the existence question unresolved.