Hittitologist Alwin Kloekhorst has argued that the ḫi-conjugation of Hittite is virtually the same as the reconstructed perfect (or stative) of Proto-Indo-European, but without any reduplication.
Deriving the Hittite ḫi-conjugation directly from the PIE thematic presents (like Couvreur did) involves huge problems, but that there is some connection seems very probable (in line with Pedersen and Watkins).
All these three solutions, according to Jasanoff, present massive problems, not the least of which being that the way verbs are distributed between the two Hittite conjugations seems arbitrary, by and large, compared to the other PIE daughter languages.
Even for this reason, it seems more probable that two categories serving one function have largely merged in the other languages (Sanskrit, Greek, Latin, etc.).
The plural is omitted altogether because it presents significantly greater problems than the singular, as the "normal" forms were generalized from the Hittite mi-conjugation to the ḫi-conjugation, probably by analogy.)