  The flux power spectrum of the high resolution Lyman-$\alpha$ forest data
exhibits suppression at small scales. The origin of this suppression can be due
to long-sought warm dark matter (WDM) or to thermal effects, related to the
largely unknown reionization history of the Universe. Previous works explored a
specific class of reionization histories that exhibit sufficiently strong
thermal supression and leave little room for warm dark matter interpretation.
In this work we choose a different class of reionization histories, fully
compatible with available data on evolution of reionization, but much colder
then the reionization histories used by previous authors in determining the
nature of dark matter, thus leaving the broadest room for the WDM
interpretation of the suppression in the flux power spectrum. We find that WDM
thermal relics with masses below 1.9 keV (95% CL) would produce a suppression
at scales that are larger than observed maximum of the flux power spectrum,
independently of assumptions about thermal effects. This WDM mass is
significantly lower than previously claimed bounds, demonstrating the level of
systematic uncertainty of the Lyman-$\alpha$ forest method, due to the previous
modelling. We also discuss how this uncertainty may affect also data at large
scales measured by eBOSS.
