The local moduli of Sasaki-Einstein rational homology 7-spheres and invertible polynomials

We compute the dimension of the local moduli space of Sasaki-Einstein metrics for links of invertible polynomials coming from the list of Johnson and Koll\'ar of anticanonically embedded Fano 3-folds of index 1 that produce rational homology 7-spheres, that is, 7-manifolds whose rational-homology equals that of the 7-sphere. In order to do so, we propose additional conditions to the Diophantine equations associated to this problem. We also find solutions for the problem associated to the moduli for the Berglund-H"ubsch duals of links arising from Thom-Sebastiani sums of chain and cycle polynomials. In particular we give an explicit description of local moduli of the complex structure for the corresponding K"ahler-Einstein orbifold which in this setting is a rational homology projective 3 -spaces with quotient singularities.