The Italian cut did have an English sound mix but it was lost?
No. That’s basically a wrong assumption of how these films were made.
They were filmed without sound recording on set, as the films were later entirely dubbed in each of the languages they were released in. For the longer Italian version, the Italian dub was recorded in Italy. For the (shorter) US version, the English dub was recorded in the USA. During filming, every actor spoke his native language.
There never was an English mix for the scenes from the Italian cut in the first place. The actors had their notes on what they spoke when they played their scenes (and as Clint Eastwood pointed out in an interview, he always kept his notes as he feared the Italians might lose them - and they surely did).