This incorporates blueblimp's idea of adding in Mandarin.
Have four non-treasure cards (or fewer if one is a bridge, highway, or princess) in play while having a hand with three horns of plenty and a watchtower. Necropolis may help set this up more quickly. Suppose the Cultists, Duchies, and remaining HoPs are untouched and at least four Mandarins are available.
HoP -- gain a HoP, top deck it [2 HoP in hand, 1 on deck]
HoP -- gain a HoP, top deck it [1 HoP in hand, 2 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (2 HoP + something) [2 HoP in hand, 0 on deck]
HoP -- gain a Mandarin, top deck all four HoPs [1 HoP in hand, 4 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all HoP) [3 HoP in hand, 1 on deck]
HoP -- gain a HoP, top deck it [2 HoP in hand, 2 on deck]
HoP -- gain a HoP, top deck it [1 HoP in hand, 3 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all HoP) [3 HoP in hand, 0 on deck]
HoP -- gain a HoP, top deck it [2 HoP in hand, 1 on deck]
HoP -- gain a HoP, top deck it [1 HoP in hand, 2 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (2 HoP + something) [2 HoP in hand, 0 deck]
HoP -- gain a Mandarin, top deck all eight HoPs [1 HoP in hand, 8 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all 3 HoP) [3 HoP in hand, 5 on deck]
HoP -- gain a Duchy, trash HoP [2 HoP in hand, 5 on deck]
HoP -- gain a Duchy, trash HoP [1 HoP in hand, 5 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all 3 HoP) [3 HoP in hand, 2 on deck]
HoP -- gain a Duchy, trash HoP [2 HoP in hand, 2 on deck]
HoP -- gain a Duchy, trash HoP [1 HoP in hand, 2 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (2 HoP + something) [2 HoP in hand, 0 on deck]
HoP -- gain a Mandarin, top deck all four HoPs [1 HoP in hand, 4 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all 3 HoP) [3 HoP in hand, 1 on deck]
HoP -- gain a HoP, top deck it [2 HoP in hand, 2 on deck] *HoP pile is empty
HoP -- gain a Mandarin, top deck all three HoPs[1 HoP in hand, 5 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (all 3 HoP) [3 HoP in hand, 2 on deck]
HoP -- gain a Duchy, trash HoP [2 HoP in hand, 2 on deck]
HoP -- gain a Duchy, trash HoP [1 HoP in hand, 2 on deck]
HoP -- gain a Cultist, trash it, draw 3 cards (2 HoP + something) [2 HoP in hand, 0 on deck] *Cultist pile is empty
HoP -- gain a Duchy, trash HoP [1 HoP in hand, 0 on deck]
HoP -- gain a Duchy, trash HoP [0 HoP in hand, 0 on deck] *Duchy pile is empty
This empties the HoP, Cultist, and Duchy piles in one go.
If one HoP has been purchased by your opponent you should still be able to modify this to empty these piles in one go.
If two HoPs have been purchased, you should be able to get 7 Duchies from HoPs, and then given the spare draws from the Cultists when only 2 HoPs were on top of your deck, you could probably just buy the eighth Duchy.
Missing Cultists is a bigger problem, but you could still snatch many of the Duchies, and hopefully your Mandarins would help you reach the remaining Duchies.
OK, yeah, this would be hard to set up. Any ideas on making this easier? Maybe opening Mandarin/Mint, then using the Mandarin with the remaining coppers to buy a Horn of Plenty which is then minted. Your four non treasure cards could be Necropolis, Mandarin, Mint and some cantrip.