Comment on 'Product states and Schmidt rank of mutually unbiased bases in dimension six'

A lemma by Chen et al. [J. Phys. A: Math. Theor. 50, 475304 (2017)] provides a necessary condition on the structure of any complex Hadamard matrix in a set of four mutually unbiased bases in ${\mathbb{C}}^6$. The proof of the lemma is shown to contain a mistake, ultimately invalidating three theorems derived in later publications.