The six-multiplication maximum-degree conjecture

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six matrix-matrix multiplications. Six-multiplication conjecture. For complex coefficients,

$$\max\{d: \Pi_d \subset \overline{\Pi^*_{2^6}}\}=32.$$

For real coefficients,

$$\max\{d: \Pi_d \subset \overline{\Pi^*_{2^6}}\}=30.$$

The values are supported by numerical constructions: the paper reports real and complex solutions, while the asserted maximality remains unproved.