Strict convexity characterization by linear extension of sphere embeddings

Let $X$ be a normed space with unit sphere $S_X$, and let $Y$ range over normed spaces with unit spheres $S_Y$. An isometric embedding $S_Y\to S_X$ is a distance-preserving map between the unit spheres. Strict-convexity characterization conjecture. The space $X$ is strictly convex if and only if every isometric embedding $S_Y\to S_X$ extends linearly. The surrounding discussion motivates this as a proposed characterization: strict convexity is known there to force affine behavior for isometric embeddings of whole normed spaces, but the sphere version is presented as an open direction.