Time Reloader on flipping Mimighouls

[Oreios]

Time Reloader has the following effect:
"During the Main Phase, if a monster on the field is flipped face-up: You can Special Summon this card from your hand, then change 1 face-up monster on the field to face-down Defense Position."

Mimighouls are flip monsters that can summon themselves face-down onto the opponents field and upon flip triggers 2 effects:
- Something beneficial for you/not beneficial for the opponent
- Switch control of the card.

If my opponent Flip summons a Mimighoul, triggers its flip effect and I chain Time Reloader to summon itself and book the flipped mimighoul, does the mimighoul return to my field face-down or does it stay face-down on my opponents field?