The dimension conjecture for unreduced evaluation schemes with $2^m$ multiplications

Let $m>2$ and let $\Pi^*_{2^m}$ denote the polynomial set associated with evaluation schemes using $2^m$ matrix-matrix multiplications. Numerical computations suggest that generic unreduced schemes give the sharp parameter count. Dimension conjecture.

$$\dim(\Pi^*_{2^m})=m^2.$$

The preceding argument proves only the upper bound $\dim(\Pi^*_{2^m})\leq m^2$; equality was verified numerically for $m\leq 10$, but no general proof is supplied.