Ferrofluid Fountain: Perpetual Motion Machine?

On 2023-04-28 the post “How Ferrofluid Climbs Chains of Magnets” discussed how and why ferrofluid can climb chains of magnets, seemingly defying gravity. Can you, then, make a perpetual motion machine by drawing the ferrofluid up with magnets, then letting it fall back into a beaker under the force of gravity?

Doooh! Stupid conservation of energy!”

But why, exactly doesn’t it work?