The MUB maximum-moment conjecture

Let $X$ be a normalized uniformly distributed random vector. For mutually unbiased bases, let $m_1,m_2,\ldots,m_k$ denote the relevant maximum second-moment quantities, and let $u_1,u_2,\ldots,u_k$ denote the corresponding quantities for the comparison random variables. MUB maximum-moment conjecture.

$$E(\max(m_1,m_2,\ldots,m_k))\geq E(\max(u_1,u_2,\ldots,u_k)).$$

This conjecture proposes that mutually unbiased bases globally support at least as large an expected maximum second moment as the corresponding comparison variables. The surrounding discussion gives asymptotic motivation but does not establish the asserted inequality.