We show that any two same-genus, oriented, boundary-parallel surfaces bounded by a non-split, alternating link in the 4-ball are smoothly isotopic fixing the boundary. In other words, any same-genus Seifert surfaces for a non-split, alternating link become smoothly isotopic fixing the boundary once their interiors are pushed into the 4-ball. As a consequence, we conclude that a smooth surface in $S^4$ obtained by gluing two Seifert surfaces for a non-split, alternating link is always smoothly unknotted. This work is joint with Seungwon Kim and Maggie Miller.
