I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of genus less than $g$?

If $g_n(K)$ is the smallest possible genus of an embedded surface bounding the lift of $K$ in its $n$-fold cyclic branched cover, what happens to $g_n(K)$ if we allow $n$ to grow? (Assume $K$ is not fibered.)