Flenner–Zaidenberg's weak rigidity conjecture

Let $C\subset\mathbb P^2$ be a reduced, irreducible rational cuspidal curve, 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 $(V,D)$. Write $\chi(T_V\langle D\rangle)$ for the Euler characteristic of this logarithmic tangent bundle.

Weak rigidity conjecture. If $C$ has at least three singularities, then

$$\chi(T_V\langle D\rangle)=0.$$

This is presented as another conjecture due to Flenner and Zaidenberg. The supplied text gives no general resolution.