Lifetimes of low-lying excited states in the $\nu i_{13/2^{+}}$ bands of the neutron-deficient osmium isotopes $^{169,171,173}$Os have been measured for the first time using the recoil-distance Doppler shift and recoil-isomer tagging techniques. An unusually low value is observed for the ratio $B(E2;21/2^{+}\to 17/2^{+})/B(E2;17/2^{+}\to 13/2^{+})$ in $^{169}$Os, similar to the “anomalously” low values of the ratio $B(E2;4_1^{+}\to 2_1^{+})/B(E2;2_1^{+}\to 0_{\mathrm{gs}}^{+})$ previously observed in several transitional rare-earth nuclides with even numbers of neutrons and protons, including the neighbouring $^{168,170}$Os. Furthermore, the evolution of $B(E2;21/2^{+}\to 17/2^{+})/B(E2;17/2^{+}\to 13/2^{+})$ with increasing neutron number in the odd-mass isotopic chain $^{169,171,173}$Os is observed to follow the same trend as observed previously in the even-even Os isotopes. These findings indicate that the possible quantum phase transition from a seniority conserving structure to a collective regime as a function of neutron number suggested for the even-even systems is maintained in these odd-mass osmium nuclei, with the odd valence neutron merely acting as a “spectator”. As for the even-even nuclei, the phenomenon is highly unexpected for nuclei that are not situated near closed shells.