The weak rigidity conjecture for rational cuspidal curves

Let $\bar E\subseteq\mathbb{P}^2$ be a rational cuspidal curve of log general type, and let $\pi:(X,D)\to(\mathbb{P}^2,\bar E)$ be the minimal log resolution of singularities. Let $\mathcal{T}_X(-\log D)$ denote the logarithmic tangent sheaf of $X$ along $D$. Flenner–Zaidenberg's weak rigidity conjecture. The Euler characteristic of $\mathcal{T}_X(-\log D)$ vanishes. Equivalently, $K_X\cdot(K_X+D)=0$, or $p_2(\mathbb{P}^2,\bar E)=0$. The conjecture concerns rigidity properties of log-general-type rational cuspidal curves; the source gives no evidence of resolution.