I had earlier posted a "proof" of this here in 3D and 4D. It's wrong. Then Fred Lunnon, whom I thought (due to miscommunication) had agreed with my proof for over a month -- surprised me by posting his paper proving this on the arXiv, and not crediting my earlier proof. However, the explanation is that Lunnon actually had detected that my "proofs" were bogus due to an exceedingly stupid error. This was revealed just now when I enquired of Lunnon. The error was, I had "E" and I had "new E" and got the two confused. Very dumb, but I somehow managed it. So anyhow, I retract my proofs. Even earlier, Lunnon himself had sent me email containing his own bogus proof, which I had pointed out the bogusness in by return email. Fortunately in that case, there was no miscommunication, i.e. we both realized the proof was wrong. To my credit, I then produced a new proof, but to my discredit, it was wrong. Then Lunnon has now produced a third proof, to his credit -- in 3D -- which hopefully is now right! Lunnon's new proof employs quaternion GCDs, which was an approach I had originally suggested and pushed to Lunnon... although in my bogo-proofs I had wrongly thought the need for them had gone away. I'm sorry this all happened and hope his latest proof is correct. As far as I currently know my other proofs about other Heronian simplex topics remain ok, and nobody has pointed out any errors or claimed-errors in them... yet...