For example suppose the sprayer was a metal through which hydrogen can diffuse, perhaps sucked in osmotically, with the OH somehow stuck at that point on the metal surface, then over time all the surface "pores" would be clogged, but the bulk metal still osmotically sucks hydrogen, so perhaps the OH's on the surface are forced to recombine into H2O2 to make space. (I think their explanation is much more likely than mine, but it illustrates the importance of tracing all the chemical end products, to make sure the reaction is happening where one thinks it is happening. essentially: perform the experiment again in a continuous mode, in a closed recipient while monitoring hydrogen concentration. Continuous mode to prevent hydrogen gas diffused in metal components to be outgassed by say temperature effects of thermal cycling)
I think their explanation is very likely though. That the concentrations were higher in smaller droplets is a strong indicator that this is a surface effect (since volume of a droplet scales down with the third power in radius, while surface area scales down only with a second power in radius). Then remember that water is a polar molecule, and if they are anything like magnets, they don't like being parallel, so I would expect the electric dipole of a water molecule to try to lay in the plane of the surface on average, but as we all know, you cant comb a hairy ball!
But I am not a chemist.
In this case they get H2O2 (neutral) so the H must be neutral, so they must combine to form H2 (neutral). (Perhaps the H radical can survive some time in water until it combine with other H and form the H2.)
[1] A small technical detail is that the reaction is 2 * H2O --> H3O+ + HO-, not H2O --> H+ + HO- , but H+ is the usual shorthand.