The immersed-genus extension of Property G

Let $Y$ be a closed, oriented, connected $3$--manifold, let $K\subset Y$ be a null-homologous knot, and let $F$ be a taut Seifert surface for $K$. Set

$$X=(Y\times[0,1])\#N(\mathbb CP^2\#\overline{\mathbb CP^2}).$$

Let $\iota\colon (Y,K)\to (X,K\times\{0\})$ be the inclusion map. The immersed-genus extension of Property G. If there is a properly embedded smooth connected surface $G\subset X$ with boundary $K\times\{0\}$, homology class $[G]=\iota_*[F]\in H_2(X,K\times\{0\})$, and genus $g(G)<g(F)$, then $K$ has Property G.