The seven-multiplication maximum-degree conjecture

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with seven matrix-matrix multiplications. Seven-multiplication conjecture. In the complex-coefficient case,

$$\max\{d: \Pi_d \subset \overline{\Pi^*_{2^7}}\}=42.$$

The value is inferred from a numerically computed solution for the Taylor expansion of the exponential using a structured scheme; the numerical evidence does not establish that degree $42$ is maximal.