Banach's affine section conjecture for convex bodies

Let $V$ be a finite-dimensional real vector space, let $2\leq k<\dim V$, and let $K\subset V$ be a convex body. Suppose that all $k$-dimensional sections of $K$ through a fixed interior point are affinely equivalent. Banach's affine section conjecture. Then $K$ is an ellipsoid. This is stated in the paper as an equivalent formulation of Banach's isometric conjecture, with the quantitative affine version serving as its stability analogue; the exact assertion remains open.