Nonpositivity conjecture for the logarithmic tangent sheaf of cuspidal curves on Hirzebruch surfaces

Let $C$ be a rational cuspidal curve with four or more cusps on a Hirzebruch surface $\mathbb{F}_e$. Let $(V,D)$ be the minimal embedded resolution of $C$, and let $\Theta_V\langle D\rangle$ be the logarithmic tangent sheaf. Logarithmic Euler-characteristic conjecture.

$$\chi(\Theta_V\langle D\rangle)\leq 0.$$

The inequality is motivated by the table of constructed examples, for which the Euler characteristic is nonpositive; the source gives no proof for all rational cuspidal curves with four or more cusps.