Dimca–Sticlaru conjecture on branches of singularities of free and nearly free curves

Let $C\subset\mathbb{P}^2$ be an irreducible plane curve. Using the Jacobian algebra and the quotient $N(f)=I_f/J_f$ associated with a homogeneous defining equation $f$, call $C$ free when $N(f)=0$ and nearly free when $N(f)\ne 0$ and $\dim N(f)_k\leq 1$ for every $k$. A singularity has at most $r$ branches if its local reduced germ has no more than $r$ irreducible local components.

Dimca–Sticlaru's branch 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.

The conjecture gives local restrictions on singularities of irreducible free and nearly free plane curves. The cited passage does not report a resolution, so both assertions are recorded as open.