Zauner's conjecture on maximal complex equiangular tight frames

Let $M\geq 2$. An equiangular tight frame (ETF) in $\mathbb{C}^M$ with $N$ vectors is denoted by $\operatorname{ETF}(M,N)$; the maximal possible number of vectors in the complex case is $N=M^2$. Zauner's conjecture. There exists an equiangular tight frame $\operatorname{ETF}(M,M^2)$ for every $M\geq 2$. These maximal ETFs are also called symmetric, informationally complete, positive operator-valued measures (SIC-POVMs) and are important in quantum Bayesianism, quantum state tomography, and quantum cryptography. Their existence is conjectured in every dimension, but the general existence question remains open.