Degree-and-type conjecture for representations of code loops

Let $L$ be a code loop, and let $V_{1}$ and $V_{2}$ be representations of $L$. The degree and type of a representation are the corresponding representation invariants.

Degree-and-type conjecture. If $V_{1}$ and $V_{2}$ have the same degree and type, then they are isomorphic.

The claim is motivated by the fact that, for code loops of ranks $3$ and $4$, the type determines the code loop up to isomorphism. The supplied text gives no resolution beyond suggesting that the rank-$3$ and rank-$4$ observation may hold in general.