physics engine – Collision response using the Minimum Translation Vector

I am trying to use the Minimum Translation Vector (MTV) returned from the SAT algorithm for simple collision response. Please see the example below.

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By separating the colliding boxes by moving the black box along the MTV, the box is now colliding with the other green box below. Moving the black box along the MTV for the collision with this new box then brings the box back to colliding with the first box, and an infinite cycle of trying to separate the boxes begins.

Am I missing something, or is collision response not possible using the MTV?