We learned in tSR that Aielmen currently sing only when going into battle, and for funerals (Rhuarc stated this when the Taardad were singing the "Wash the Spears" song). But we also know that (a) Loial will be singing to the apple trees to honor Perrin's slain kinfolk, (b) Gaul, Chiad and Bain will be there to accompany Perrin and Faile, (c) Aram the Tinker is likely to be there too since he is sticking to Perrin as if Perrin is Aram's lifesaver, and therefore (d) all the necessary people and conditions are set for the song to be found.
I feel sure that Gaul, Chiad and Bain will see this as a funeral, and thus will sing. And from the fact that many laments in our world are wordless (e.g. most of the mid-east ones I have heard) even if they are heart-felt expressions, the Aiel song has a high probability of fitting in with Loial's to form the "true?" song form that Rand learned of in Rhuidean via his ancestors' memories.
Even though there is no Nym present to braid the song into a full field-enrichening one, as Rand "heard," it should be noted that only the Ogier and the Aiel actually sang even back at the breaking (at least as far as we know). Further, Loial also saved at least a part of the Green Man (as that final Nym was called) in tEotW, to the degree that the blight was again held back by that oak. So there is a reasonable chance that the Nym may be recovered as a race as well.
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