Di-extensions in monoids and unital magmas

Decide whether, in each of the categories Mag of unital magmas and Mnd of monoids, every short exact sequence of short exact sequences is a di-extension.

Background

Di-extensions (3×3-diagrams with exact rows and columns) are central to the homological framework. While they hold broadly in di-exact settings, it is unclear whether categories like monoids and unital magmas satisfy the di-extension property for all short exact sequences of short exact sequences.