  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 $10$.
