Multidimensional non-martingale for which linear combinations of its components are all martingales

Question

Does there exist an $\mathbb{R}^d$-valued stochastic process $(X_n)_{n \geq 0}$ (discrete time/continuous time) such that for all $a=(a_1,...,a_d) \in \mathbb{R}^d$, $\langle a,X_n \rangle = a_1 X^{(1)}_n + ... + a_d X^{(d)}_n$ is a (one-dimensional) martingale (in its own filtration), but $X$ is not a martingale in its own filtration?