Is it possible to bind a conjured elemental for an extended duration?

I am curious about the interaction between the spells Conjure Elemental and Planar Binding (plus Magic Circle to avoid the elemental going berserk).

Assume a Warlock of 9th level or higher with the Minion of Chaos invocation who knows Magic Circle and Planar Binding.

Start by casting Magic Circle at 5th level, so it lasts for three hours. Take a short rest, now the circle is down to two hours. Use Minion of Chaos to cast Conjure Elemental within the circle, then immediately drop concentration (losing control of the elemental) and begin casting Planar Binding. After an hour, Planar Binding cast is complete and the elemental is theoretically bound.

Does this work as described?