A transversal logical gate for an $ n $ qubit code is a gate from the group of local unitaries $$ \bigotimes_{i=1}^n U(2) $$ which also preserves the codespace. For an $ ((n,K,d)) $ code we say a particular logical gate $ \tilde{g} \in U(K) $ can be implemented transversally if there exists some gate $$ g=\bigotimes_{i=1}^n g_i \in \bigotimes_{i=1}^n U(2) $$ which implements $ \tilde{g} $ when restricted to the codespace. If all the $ g_i $ are equal then we say $ \tilde{g} $ is strongly transversal.
My question is the following: For every $ [[n,1,d]] $ code does there exist some basis for the codespace with respect to which logical $ X $ and logical $ Z $ have a strongly transversal implementation?