Flenner–Zaidenberg's rigidity conjecture for rational cuspidal curves

Let $C\subset\mathbb P^2$ be a reduced, irreducible rational cuspidal curve, and let $V\to\mathbb P^2$ be the minimal embedded resolution of its singularities. Let $D$ be the simple normal-crossings total transform of $C$, and let $T_V\langle D\rangle$ be the logarithmic tangent bundle of the pair $(V,D)$. Define FZ-projectively rigid by $h^1(T_V\langle D\rangle)=0$ and FZ-unobstructed by $h^2(T_V\langle D\rangle)=0$. Also, $es$-rigidity is the corresponding equisingular rigidity condition.

Rigidity conjecture. If $C$ has at least three singularities, then $C$ is FZ-projectively rigid and FZ-unobstructed. In particular, $C$ is $es$-rigid.

The source attributes this conjecture to Flenner and Zaidenberg and states that it has been checked for the families ${\mathcal F}{\mathcal Z}_1(d,a)$ and ${\mathcal F}{\mathcal Z}_2(k)$.