The possible existence of warm ($T_{rm eff}sim19,000$ K) pulsating DA white dwarf (WD) stars, hotter than ZZ Ceti stars, was predicted in theoretical studies more than 30 yr ago. However, to date, no pulsating warm DA WD has been discovered. We re-examine the pulsational predictions for such WDs on the basis of new full evolutionary sequences. We analyze all the warm DAs observed by TESS satellite up to Sector 9 in order to search for the possible pulsational signal. We compute WD evolutionary sequences with H content in the range $-14.5 lesssim log(M_{rm H}/M_{star}) lesssim -10$, appropriate for the study of warm DA WDs. We use a new full-implicit treatment of time-dependent element diffusion. Non-adiabatic pulsations were computed in the effective temperature range of $30,000-10,000$ K, focusing on $ell= 1$ $g$ modes with periods in the range $50-1500$ s. We find that extended He/H transition zones inhibit the excitation of $g$ modes due to partial ionization of He below the H envelope, and only in the case that the H/He transition is assumed much more abrupt, models do exhibit pulsational instability. In this case, instabilities are found only in WD models with H envelopes in the range of $-14.5 lesssim log(M_{rm H}/M_{star}) lesssim -10$ and at effective temperatures higher than those typical of ZZ Ceti stars, in agreement with previous studies. None of the 36 warm DAs observed so far by TESS satellite are found to pulsate. Our study suggests that the non-detection of pulsating warm DAs, if WDs with very thin H envelopes do exist, could be attributed to the presence of a smooth and extended H/He transition zone. This could be considered as an indirect proof that element diffusion indeed operates in the interior of WDs.
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