The branch-number conjecture for free and nearly free plane curves

Let $C$ be an irreducible plane curve. A singularity of $C$ has a branch for each local analytic branch, and $C$ is free or nearly free according to the corresponding classes of plane divisors introduced in the paper.

Branch-number conjecture. (i) Any free irreducible plane curve $C$ has only singularities with at most two branches. (ii) Any nearly free irreducible plane curve $C$ has only singularities with at most three branches.

This is stated as a related conjectural property of irreducible free and nearly free curves. The supplied text gives no resolution status or partial result for these branch bounds.