Bound the number of lines containing all multiple points

Prove that for every line arrangement A in the complex projective plane P2 with first exponent d1, if N is the minimum number of lines whose union contains all multiple points of A, then N is at most d1 + 1.

Background

The paper proves in Corollary 1.13 that all multiple points of a line arrangement A are contained in a curve of degree at most d1 + 1, where d1 is the first exponent of A. The authors observe that the curve produced by their argument contains at least mp lines, but they seek the stronger assertion that all multiple points can always be covered by a union of at most d1 + 1 lines.

The conjecture is supported by Corollary 1.15 and Theorem 1.16, which establish it for arrangements containing a point of multiplicity d1 and for several important classes, including arrangements with only double points, certain arrangements with triple points, arrangements having a point of multiplicity at least half the number of lines, and supersolvable arrangements. The authors further explain that the conjecture is equivalent to requiring the polynomial gp associated with at least one multiple point to factor completely into linear factors.