Zauner's weak conjecture on symmetric informationally complete measurements

For each positive integer $d\geq 2$, a SIC POVM is a set of $d^2$ rank-one positive operators on $\mathbb{C}^d$ that forms an informationally complete symmetric positive operator-valued measure; equivalently, unit vectors $w_i\in\mathbb{C}^d$ generate a SIC POVM when $|\langle w_i,w_j\rangle|^2=\frac{1}{d+1}$ for all $i\ne j$. Zauner's weak conjecture. For every positive integer $d\geq 2$, there exist $d^2$ unit vectors $\{w_i\}_{i=1}^{d^2}\subseteq\mathbb{C}^d$ such that

$$|\langle w_i,w_j\rangle|^2=\frac{1}{d+1}$$

for all $i\ne j$. This is the SIC POVM existence problem. It remains open in general, although concrete solutions are known in many dimensions, including $d=1$ through $21$, $24$, $28$, $30$, $31$, and $35$.