(Megszokottól eltérő időpont!) We show that $ 2^d \to 1 $ quantum random access codes are optimised uniquely by measurements corresponding to mutually unbiased bases. Therefore, they provide an ideal self-test in the prepare-and-measure scenario for mutually unbiasedness in arbitrary dimension -- that is, we can characterise the measurements based solely on their outcome statistics. In dimensions where there is only one equivalence class of pairs of mutually unbiased bases, this characterisation is up to a unitary transformation and a global complex conjugation. Moreover, they self-test the states used on the encoding side in the same manner. While proving this result, we also provide a necessary condition for saturating an operator norm-inequality derived by Kittaneh.