Minimal defining identities conjecture for the variety of code loops

Let ${\cal E}$ be the variety of Moufang loops generated by all code loops, defined by the identities

$$x^4=1,\qquad [x,y]^2=1,\qquad (x^2,y)=1,\qquad (x^2,y,z)=1.$$

Minimal identities conjecture. These identities form a minimal set of identities defining the variety ${\cal E}$.

The identities originally used to define ${\cal E}$ are noted to be nonindependent; the conjecture asks whether the four displayed identities are sufficient and minimal.