We present deeply virtual π^{0} electroproduction cross-section measurements at x_{B}=0.36 and three different Q^{2} values ranging from 1.5 to 2 GeV^{2}, obtained from Jefferson Lab Hall A experiment E07-007. The Rosenbluth technique is used to separate the longitudinal and transverse responses. Results demonstrate that the cross section is dominated by its transverse component and, thus, is far from the asymptotic limit predicted by perturbative quantum chromodynamics. Nonetheless, an indication of a nonzero longitudinal contribution is provided by the measured interference term σ_{LT}. Results are compared with several models based on the leading-twist approach of generalized parton distributions (GPDs). In particular, a fair agreement is obtained with models in which the scattering amplitude includes convolution terms of chiral-odd (transversity) GPDs of the nucleon with the twist-3 pion distribution amplitude. This experiment, together with previous extensive unseparated measurements, provides strong support to the exciting idea that transversity GPDs can be accessed via neutral pion electroproduction in the high-Q^{2} regime.