Cube of Force(option 4) does not stop Banishment.
snippets of Cube of Force
Each face has a different effect....a barrier of Invisible force springs into existence, forming a cube 15 feet on a side. The barrier is centered on you, moves with you, and lasts for 1 minute....
face - charges
4 — 4 — Spell effects can 't pass through the barrier.
Nothing in the item's description implies that spells can't be cast from within the cube (compared to AntiMagic Field). Banishment doesn't "send out" an effect, so there's no reason that the two should interact.
The archmagi would walk in and cast Banishment. Pending a failed save, the player who initiated the Cube of Force would blink out of existence, along with their cube effect. Then 1 minute later, they would reappear, without the cubes effect, as more than 1 minute has passed since the Cube of Force was activated.
PS - Though this doesn't apply to this exact interaction, I'm not sure if you could teleport from inside of the cube to outside or vice versa (think Dimension Door). My inclination is to say, that you could, as you're bypassing the barrier. Much as if it were a physical wall. Wizards can't walk through walls, but they can teleport to the other side.