The SLP conjecture for Hartshorne–Rao modules of general unions of lines

Let $C\subset\mathbb P^3$ be the union of $r$ general lines. The strong Lefschetz property conjecture. Then $M(C)$ has SLP.

Here $M(C)$ denotes the Hartshorne–Rao module of $C$, and SLP means the strong Lefschetz property. The conjecture extends the verified cases, where multiplication by powers of a general linear form has maximal rank through the third power, to all powers.