Laplacian estimate for the Chen–LeBrun–Weber extremal Kähler metric

Let $k$ be the extremal Kähler metric on $\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$ such that $s^{-2}k$ is the Chen–LeBrun–Weber metric with Einstein constant equal to $3$, where $s$ denotes the scalar curvature of $k$. Laplacian estimate conjecture. The pointwise estimate

$$\Delta s^2<\frac{15}{4}$$

holds. Strong numerical evidence is given for this estimate, and it would suffice to establish the preceding instability results for the Chen–LeBrun–Weber examples.