Multiplication identities of Cayley-Dickson algebras after the sedenions

Question

Consider the multiplication reducts $\{*\}$ of Cayley-Dickson algebras $(X;+,-,*,0,1)$ over the real numbers $\mathbb{R}$. In both the real numbers and the complex numbers, multiplication satisfies both the commutative and associative identities. In the quaternions, multiplication satisfies the associative identity but not the commutative identity. In the octonions, multiplication satisfies neither the associative identity nor the commutative identity, but it does satisfy the alternative identity. In the sedenions, multiplication does not satisfy even the alternative identity. My question is, is the equational theory of the multiplication reducts of any Cayley-Dickson algebra past the sedenions the same as the equational theory of the multiplication reduct of the sedenions? I conjecture that it is, and I further conjecture that the equational theory is axiomatized by the flexible identity along with an infinite set of equations that collectively axiomatize power-associativity. Is this true? Also, could this possibly be an open problem?