Richard Westwood-Brookes in "Elgar and the press" (which I reviewed some years ago for this site) quotes from a review of the first performnance of the suite on 21 April 1896, from which it is clear that this was given with orchestral accompaniment - it followed a performance of Beethoven's Egmont Overture (the review does not state who conducted, but it does not appear to have been the composer). It would seem therefore that the piano version was origially intended for rehearsal purposes only, and as such would not have existed in a four-handed version. This may not resolve the mystery but it does possibly shed some further light on the matter.