Wobbling motion as an exotic collective mode in nuclei without axial symmetry, was intensively discussed during the last few years. The observation of the newly proposed transverse wobbling, first reported in $^{135}$Pr and soon after in nuclei from other mass regions, was considered as a significant discovery in low-spin nuclear structure. However, both the reported experimental results and the proposed theoretical models were lively questioned in a series of works devoted to the investigation of the low-spin wobbling mode in the same nucleus. The electromagnetic properties of the $\Delta I=1$ transitions connecting the one- to zero-phonon and the two- to one-phonon wobbling bands in $^{135}$Pr were recently remeasured, demonstrating their predominant $M1$ magnetic character which is in contradiction with the wobbling motion interpretation. These new experimental observables being well reproduced by both quasiparticle-plus-triaxial-rotor model and interacting boson-fermion model calculations, are against the previously proposed wobbling characteristics of the low-spin bands in $^{135}$Pr. On the other hand, we obtained conclusive experimental evidence for the theoretically suggested transverse wobbling bands at medium spin in $^{136}$Nd. The comparison of the experimental data with calculations utilizing the triaxial projected shell model and a new developed particle-rotor model with frozen orthogonal geometry of the active nucleons, supports the description in terms of transverse wobbling of medium-spin bands in triaxial even-even nuclei.