Walther-bound equality conjecture for essential line arrangements

Let $C:f=0$ be an essential arrangement of $d\geq 3$ lines in $\mathbb{P}^2$, and let $m(C)$ be the maximal multiplicity of an intersection point on $C$. The invariants $\nu(C)$ and $\nu'(C)$ satisfy

Walther-bound equality conjecture.

$$\nu(C)=\nu'(C)$$

if and only if either $m(C)=d-1$, or $m(C)\leq 3$.

Here $\nu'(C)$ is the upper bound for the freeness defect supplied by Walther's result. The conjecture asserts that equality occurs only for the two stated extreme types of essential line arrangements.