  In this article, we provide a relation between the $\mu$-invariants of the
dual plus and minus Selmer groups for supersingular elliptic curves when we
ascend from the cyclotomic $\mathbb{Z}_p$-extension to a
$\mathbb{Z}_p^2$-extension over an imaginary quadratic field. Furthermore we
show that the supersingular $\mathfrak{M}_H(G)$-conjecture is equivalent to the
fact that the $\mu$-invariants doesn't change as we go up the tower.
