Exceptional Fano 3-folds from rational curves

We show exceptionality of certain families of non-quasismooth weighted hypersurfaces. In particular these admit Kähler-Einstein metrics. Our examples are produced by the monomials generating the complex deformations of orbifolds whose corresponding $S^1$-Seifert bundles are smooth rational homology 7-spheres admitting Sasaki-Einstein metrics. From our construction, it follows that these exceptional Fano hypersurfaces describe elements in the boundary of the K-moduli of $\mathbb{Q}$-Fano 3-folds.