This is an interesting question, mostly from the standpoint of how such an idea came to be.
Perhaps, some years ago, one possibility is that is isobaric counterdiffusion was recognized at some level, but without really integrating that with an accurate understanding of the offgassing characteristics of helium. Possibly one could then come up with the idea that EANx dives should not follow Trimix dives.
Isobaric counterdiffusion (resulting in inner ear bubbling) can indeed be precipitated by helium-nitrogen gas switching, usually at too deep a depth. ICD occurs when a tissue such as the inner ear is highly saturated with one gas, and then switching to a different breathing mix with a high concentration of a different gas, results in diffusion of that second gas into the confined space before the first has adequately made its way out. The result is a transient supersaturation of that compartment in terms of total gas concentration, which of course generates bubbling and hence ICD DCS symptoms which can include incapacitating vertigo among others.
Switches both from high helium to high nitrogen and high nitrogen to high helium have been implicated. (This seemingly contradictory position is legitimate in that comes from the facts that whereas Helium does offgasses more quickly than nitrogen, Nitrogen is much more soluble than Helium. High solubility allows a substance to rapidly cross membranes and other tissues which can result in local transient spikes of dissolved gas even though the overall offgassing rate of Nitrogen is slower. Depending on which factor dominates, ICD may theoretically be precipitated by switching in either direction.) Depth is a major exacerbating factor.
The current theory, however, is exemplified by Weinkes recommendation that switching from Trimix to Nitrox should be reserved for depths above 100 FSW.
A rationale for this is that given the rate of offgassing of Helium, the total dissolved gas concentration (PTC) in the inner ear will subside a shallower depths using normal decompression profiles, such that a major increase in nitrogen associated with a switch, say, to EAN50 at 70 feet is no longer likely to generate ICD.
This puts the perspective on the statement quoted by the OP. Before the FIRST dive is even completed, the ICD risk is already down to the point that using EANx is safe. By the time the next dive begins the risk is that much more remote.
In looking for other issues regarding using EAN or air on the second dive, the question is, does using a high nitrogen mix as opposed to Trimix pose any increased risk of nitrogen driven DCS, other than ICD? Well, there will be SOME residual Helium in the body compartments. Nitrogen loading on the second dive will be in proportion to the nitrogen concentrations and without regards to the presence of the residual helium. Hence the total dissolved gas pressure in the body compartments during the second dive will end up being higher than if that residual helium did not exist. However, if the first dive had been a nitrogen dive (air), the situation would be much worse. N2 off gasses more slowly than He and the residual total inert gas pressure at the start of the second dive will therefore be higher with a prior nitrogen dive than with a prior trimix dive.
A prior dive with either will lengthen the decompression requirements of the second dive but to a lesser extent if the first dive is Trimix. However the conclusion that can be drawn is that beyond the lengthened decompression requirements associated with having made a first dive, there do not appear, based on current theory, to be any special risks associated with having the first dive be trimix and the second an O2/N2 mix.
BTW, although one individuals experiences dont settle issues like this, I have in fact done 175-200 FSW dives with bottom times around 40 minutes (on trimix), followed by EAN30, 120 FSW, 45 min BT dives on several instances with no problem.