Instability conjecture for the Chen–LeBrun–Weber Ricci shrinker

Let $\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$ carry the Chen–LeBrun–Weber metric, viewed as a Ricci shrinker. Instability conjecture. The manifold

$$\mathbb{C} P^2\sharp 2\overline{\mathbb{C} P^2}$$

with the Chen–LeBrun–Weber metric is an unstable Ricci shrinker. This conclusion is supported by computations but depends on the same numerical estimate for extremal Kähler metrics used in the preceding cone-instability conjecture.