Weighted Fano varieties and infinitesimal Torelli problem

We solve the infinitesimal Torelli problem for $3$-dimensional quasi-smooth $\mathbb{Q}$-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length $3$ Gushel-Mukai chain of prime smooth Fanos of coindex $3$ and degree $10$.