  Inspired by previous work of Kusner and Bauer-Kuwert, we prove a strict
inequality between the Willmore energies of two surfaces and their connected
sum in the context of isoperimetric constraints. Building on previous work by
Keller-Mondino-Rivi\`ere, our strict inequality leads to existence of
minimisers for the isoperimetric constrained Willmore problem in every genus,
provided the minimal energy lies strictly below $8\pi$. Besides the geometric
interest, such a minimisation problem has been studied in the literature as a
simplified model in the theory of lipid bilayer cell membranes.
