The B4/2 ratio of reduced transition strengths between the low-lying yrast states is systematically examined in even-even, as well as in odd-A nuclei in the 28 <= Z <= 92 region. In all standard collective models in which the structure evolves from a vibrational to an axially deformed idealized case, this ratio is always greater than unity. By examining available experimental data, this study shows that not all isotopes in the nuclear chart exhibit this characteristic. A thorough search for this anomalous behavior revealed that 41 nuclei have a B4/2 <1 ratio, with 27 isotopes having upper uncertainties that do not exceed unity. While this phenomenon has remained a mystery for a long time, in recent years it has been speculated that triaxiality could be responsible for this behavior. In this paper, the role of triaxiality is analyzed for all these nuclei with the rigid triaxial rotor and the proxy-SU(3) theoretical models. We find that triaxiality is not the only ingredient necessary for a B4/2 <1 value, but the appearance of these regions is strongly correlated with the border of a shape coexistence region, along with the presence of enhanced triaxiality.