Suppose that $E/\mathbf{Q}$ is an elliptic curve and $K$ is an imaginary quadratic field. Let $\mathbf{Q}_{\infty}$ denote the cyclotomic $\mathbf{Z}_p$ extension of $\mathbf{Q}$, and let $K_{\infty}$ denote the cyclotomic $\mathbf{Z}_p$ extension of $K$,

If we know that the $\mu$-invariant for the Selmer over $K_{\infty}$ vanishes, does it follow that the $\mu$-invariant for the Selmer over $\mathbf{Q}_{\infty}$ vanishes? That is, if we know $\mu=0$ for the bigger field $K_{\infty}$, can we "propogate it down" to show that $\mu=0$ for the field $\mathbf{Q}_{\infty}$?