A noncommutatively rigid Fano fivefold

Here's a question I have had for a while, ever since reading Noncommutative rigidity of the moduli stack of stable pointed curves by Shinnosuke Okawa and Taro Sano. They prove the cool result that $\mathrm{HH}^2(\overline{\mathcal{M}}_{g,n})=0$ (except when $(g,n)=(0,5)$, and possibly when $(g,n)=(4, 0), (3, 1), (3, 0), (2, 2), (2, 1), (2, 0), (1, 3), (1, 2)$). Here $\mathrm{HH}^2$ is the second Hochschild cohomology, and it measures the deformations of the category of coherent sheaves. This is the noncommutative rigidity in the title of their paper. Note that they are considering a smooth projective Deligne–Mumford stack whenever $g\geq 1$. Finding noncommutatively rigid varieties Because I like Fano varieties so much, I was wondering whether we can find a Fano variety $X$ for which the second Hochschild cohomology vanishes, i.e., does there exist a noncommutatively rigid Fano variety? Note that it is in fact easy to find a non-Fano example of a noncommutatively rigid smooth…