The irregular set for maps with almost weak specification property has full metric mean dimension

Let $(X,d)$ be a compact metric space, $f:X\to X$ be a continuous transformation with the almost weak specification property and $\phi :X\to \mathbb{R}$ be a continuous function. We consider the set (called the irregular set for $\phi$) of points for which the Birkhoff average of $\phi$ does not exist and show that this set is either empty or carries full Bowen upper and lower metric mean dimension.