  The excited states of unstable $^{20}$O were investigated via $\gamma$-ray
spectroscopy following the $^{19}$O$(d,p)^{20}$O reaction at 8 $A$MeV. By
exploiting the Doppler Shift Attenuation Method, the lifetime of the 2$^+_2$
and 3$^+_1$ states were firmly established. From the $\gamma$-ray branching and
E2/M1 mixing ratios for transitions deexciting the 2$^+_2$ and 3$^+_1$ states,
the B(E2) and B(M1) were determined. Various chiral effective field theory
Hamiltonians, describing the nuclear properties beyond ground states, along
with a standard USDB interaction, were compared with the experimentally
obtained data. Such a comparison for a large set of $\gamma$-ray transition
probabilities with the valence space in medium similarity renormalization group
ab-initio calculations was performed for the first time in a nucleus far from
stability. It was shown that the ab-initio approaches using chiral EFT forces
are challenged by detailed high-precision spectroscopic properties of nuclei.
The reduced transition probabilities were found to be a very constraining test
of the performance of the ab-initio models.
