  For a positive integer $m$, a (positive definite integral) quadratic form is
called primitively $m$-universal if it primitively represents all quadratic
forms of rank $m$. It was proved in arXiv:2202.13573 that there are exactly
$107$ equivalence classes of primitively $1$-universal quaternary quadratic
forms. In this article, we prove that the minimal rank of primitively
$2$-universal quadratic forms is six, and there are exactly $201$ equivalence
classes of primitively $2$-universal senary quadratic forms.
