Phantom-category existence and deformation conjecture for critical blowups of Hirzebruch surfaces

Let $n$ be a nonnegative integer, set $d(n)=6+\max\\{3,n\\}$, and let $X$ be obtained by blowing up $\mathbf{F}_n$ at $d(n)$ general points. Let $\mathcal{C}_n$ be a subcategory of $\operatorname{{D^b}}(X)$, and let $\mathrm{H}^1(X,\mathcal{T}_X)$ denote the first cohomology of the tangent sheaf while $\operatorname{HH}^2(\mathcal{C}_n)$ denotes the second Hochschild cohomology of $\mathcal{C}_n$. Phantom-category existence and deformation conjecture. The category $\operatorname{{D^b}}(X)$ has a phantom subcategory $\mathcal{C}_n$ orthogonal to an exceptional collection of line bundles of maximal length. Furthermore, the map

$$\mathrm{H}^1(X,\mathcal{T}_X) \to \operatorname{HH}^2(\mathcal{C}_n)$$

is an isomorphism, so that $\mathcal{C}_n\not\simeq\mathcal{C}_m$ for $n\ne m$. This conjecture predicts distinct phantom categories at the critical number of blowups, distinguished by their Hochschild cohomology and deformation behavior; no resolution is given in the source.