Search for the population of the low-energy continuum of a tetraneutron system was performed for reactions of a 8He beam on a deuterium target. These studies are based on the data [I. A. Muzalevskii et al., Phys. Rev. C 103, 044313 (2021)] previously used for the studies of 7H and 6H in the 2H(8He, 3He)7H and 2H(8He, 4He)6H reactions. Evidence for a hump in the 4n continuum at 3.5 +/- 0.7 and 3.2 +/- 0.8 MeV was observed in the 2H(8He, 6Li)4n and 2H(8He, 3He)7H-* 3H+4n reactions, respectively. The observed statistics is quite low (6 events and up to 40 events) corresponding to very low cross sections of few microbarns or tens of microbarns. The background conditions for the 2H(8He, 6Li)4n reaction are shown to be good, favoring the physical nature of the observed events. The 2H(8He, 3He)7H-* 3H +4n process transforms to the 2H(8He, 6Li & lowast;)4n reaction in the limit of the highest 7H decay energies. The population of the low-energy region in the 4n spectrum is found to be perfectly correlated with the population of the lowest 6Li state in the 3He+3H continuum with E & lowast; = 18 MeV. Theoretical calculations of 8He in five-body alpha + 4n and of 4n in four-body hyperspherical models are presented. The 8He wave function is shown to contain strong specific correlations, which may give rise to very low-energy structures in the tetraneutron continuum in extreme-peripheral reaction scenarios.