Associative Cayley–Dickson imaginaroid extension conjecture

Let $A$ be a Cayley-Dickson imaginaroid, and suppose that the multiplication on its suspension $S{A}$ is associative, together with some further, currently unspecified, coherence conditions. Define

$$A':= A * S{A}.$$

Cayley–Dickson extension conjecture. The space $A'$ can be given the structure of a Cayley-Dickson imaginaroid; moreover, this structure is associative if $A$ is commutative.

This proposes an inductive extension of the Cayley–Dickson construction in homotopy type theory. The unspecified coherence conditions are part of the conjectural content, so the precise scope and validity of the assertion remain open.