We prove that \[ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2, \] where \(u(x,t)\) is the solution to the linear time-dependent Schr\"odinger equation on \(\mathbb{R}^2\) with initial datum \(f\), and \(X\) is the (spatial) X-ray transform on \(\mathbb{R}^2\). In particular, we identify the best constant \(C\) and show that a datum \(f\) is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions \(d\), where the X-ray transform is replaced by the \(k\)-plane transform for any \(1\leq k\leq d-1\). In the process we obtain sharp \(L^2(\mu)\) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures \(\mu\) supported on natural "co-\(k\)-planarity" sets.