Doro's conjecture on the commutative center of Moufang loops

For a Moufang loop $M$, define its commutative center and nucleus by

$$\operatorname{C}(M)=\{c\in M\mid cx=xc\ \text{for all}\ x\in M\},$$

and

$$\operatorname{Nuc}(M)=\{a\in M\mid (ax)y=a(xy)\ \text{for all}\ x,y\in M\}.$$

Doro's conjecture. If $\operatorname{Nuc}(M)=1$, then $\operatorname{C}(M)$ is normal in $M$.

The conjecture concerns whether the commutative center is forced to be normal when the nucleus is trivial. The paper shows that the commutative center is not normal in general, thereby reopening this more restricted conjecture; the supplied text does not establish whether Doro's conjecture is resolved.