I think opponents can clear out anything, especially assuming perfect shuffle luck.
6 player:
0) let p1-p5 clear Poor Houses
1) bring every other $3-pile down to 1 card.
1) Reduce deck of p6 to CCCEEE
2) p1 turn: Play Governor for draw, p1 hand GGGGGGX. p2-p5: EEEXXX, p6:EEECCC
3) p1 plays 3 Governors for remodel, let player 2-5 remodel Estate->$3 card. In the worst case, there are 10 piles, so doing this 3 times will clear out all the $3 piles. p6 trash Coppers.
4) p1 play 3 more Governors for Remodel, p6 trashes Estates.
11 plays if I counted correctly, one for draw 10 for Remodel
Edit:
You probably don't need Governor for draw, just trash one more Copper before 2), as p2-p5 trash before p6, you don't need a buffer of 3 Coppers but just 2.
That's 10 plays of Governor, I doubt that it will get much better