I asked a more general version of this question before a while ago, here: Do you lose any more equational identities when you go past sedenions?, but it didn't get any answers. So, I am asking a more specific question now.
The sedenions $\mathbb{S}$ are the result of applying the Cayley-Dickson construction to the octonions $\mathbb{O}$, and the trigintaduonions $\mathbb{T}$ are the result of applying the Cayley-Dickson construction to the sedenions. In the universal algebra sense, all the Cayley-Dickson algebras starting with the real number system $\mathbb{R}$ can be considered as $(+,-,*,0,1)$ algebraic structures, and thus it makes sense to ask which equational identities are satisfied by them.
My current question is, is there an identity which holds for the sedenions but not the trigintaduonions? I conjecture that in fact all the Cayley-Dickson algebras beyond the sedenions have the same equational identities as the sedenions themselves, but that conjecture might in fact be an open problem that has not been resolved yet. So, I am asking a more specific version of my conjecture, in the hopes that it will be easier to resolve.
