  Let $F\langle Y \cup Z, \ast \rangle$ be the free $\ast$-superalgebra over a
field $F$ of characteristic zero and let $ \Gamma^\ast_{M^{\pm}, L^{\pm}} $ be
the $T^\ast_{\mathbb{Z}_2}$-ideal generated by the set of the $\ast$-graded
Capelli polynomials
  $Cap^{(\mathbb{Z}_2, \ast)}_{M^+} [Y^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{M^-}
[Y^-,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^+} [Z^+,X]$, $Cap^{(\mathbb{Z}_2,
\ast)}_{L^-} [Z^-,X]$ alternating on $M^+$ symmetric variables of homogeneous
degree zero, on $M^-$ skew variables of homogeneous degree zero, on $L^+$
symmetric variables of homogeneous degree one and on $L^-$ skew variables of
homogeneous degree one, respectively.
  We study the asymptotic behavior of the sequence of $\ast$-graded
codimensions of $\Gamma^\ast_{M^{\pm}, L^{\pm}}.$ In particular we prove that
the $\ast$-graded codimensions of the finite dimensional simple
$\ast$-superalgebras are asymptotically equal to the $\ast$-graded codimensions
of $\Gamma^\ast_{M^{\pm}, L^{\pm}}$, for some fixed natural numbers $M^+, M^-,
L^+$ and $L^-$.
