Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$ is an immersion $f\colon S\to X$ such that $f(\partial S)$ is a longitude of $K$. We refer to the genus of $S$ as the genus of the immersed Seifert surface.

Gabai showed that the minimal genus of an immersed Seifert surface is the same as the minimal genus of an embedded Seifert surface. I was wondering whether a stronger statement holds: every immersed Seifert surface of minimal genus is an `embedded Seifert surface in disguise', i.e. homotopic (through immersions) to an embedded Seifert surface.

For higher genus I can think of immersed Seifert surfaces that certainly do not from Seifert surfaces, e.g. there exists an immersed Seifert surface $f\colon S\to X_U$ of genus one for the unknot $U$ such that there are curves on $f(S)$ which are non-trivial in $H_1(X_U)$.