By Jordan algebra I mean a finite-dimensional vector space $A$ endowed with a bilinear product satisfying $xy=yx$ and $(x^2,y,x)=0$ for every $x,y\in A$.
I was studying the paper "Power-Associative Rings of Characteristic Two" by Kokoris and by the end of the paper he proves that "Every Jordan algebra over a field of characteristic 2 with at least four elements is power-associative" but does not mention what is known about the field $\mathbb{F}_2$ (the field with only two elements).
This is a classic result when stated for $char\neq2$. I have tried to prove or find a counterexample myself but could only show that $x^5$ is well-defined. Is there a proof or disproof for this case? In fact, I couldnt even find out if this is an open problem or not. Thanks in advance.