  We study the work fluctuations in ergotropic heat engines, namely two-strokes
quantum Otto engines where the work stroke is designed to extract the ergotropy
(the maximum amount of work by a cyclic unitary evolution) from a couple of
quantum systems at canonical equilibrium at two different temperatures, whereas
the heat stroke thermalizes back the systems to their respective reservoirs. We
provide an exhaustive study for the case of two qutrits whose energy levels are
equally spaced at two different frequencies by deriving the complete work
statistics. By varying the values of temperatures and frequencies, only three
kinds of optimal unitary strokes are found: the swap operator $U_1$, an idle
swap $U_2$ (where one of the qutrits is regarded as an effective qubit), and a
non trivial permutation of energy eigenstates $U_3$, which indeed corresponds
to the composition of the two previous unitaries, namely $U_3=U_2 U_1$. While
$U_1$ and $U_2$ are Hermitian (and hence involutions), $U_3$ is not. This point
has an impact on the thermodynamic uncertainty relations (TURs) which bound the
signal-to-noise ratio of the extracted work in terms of the entropy production.
In fact, we show that all TURs derived from a strong detailed fluctuation
theorem are violated by the transformation $U_3$.
