Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:

Q. Does every cancellative duo semigroup embed into a group?

A semigroup $S$ is cancellative if, for every $a \in S$, right and left multiplication by $a$ are both injective functions on $S$; and it is duo if $aS = Sa$ for all $a\in S$. (Here, everything is written multiplicatively.)

By a famous example of Anatoly Malcev, not every cancellative semigroup is embedable into a group. So, one could also ask if Malcev's example is duo. Any reference?