Mandayam et al.'s conjecture on strongly unextendible mutually unbiased bases

A set of mutually unbiased bases (MUBs) in $\mathbb{C}^{2^m}$ is strongly $\mathbb{C}$-unextendible if no additional basis can be added even after allowing the bases to be changed by a common unitary transformation, in the sense used in the source. Mandayam et al.'s conjecture. For each integer $m>1$, there exists a set of $2^{m-1}+1$ strongly $\mathbb{C}$-unextendible MUBs in $\mathbb{C}^{2^m}$. The motivation includes known examples of $3$ strongly $\mathbb{C}$-unextendible MUBs in $\mathbb{C}^4$ and $5$ in $\mathbb{C}^8$; the conjecture proposes such a family in every dimension $2^m$ with $m>1$.