As Solís-Baltodano et al. (2021) figure out, almost a third of the total European Union budget– has been set aside for the Cohesion Policy during the 2014-2020 period. The distribution of this budget is made through three main structural and investment funds, trying to provoke the convergence in the level of development of EU countries. Specifically, the authors, by analysing this situation as a claims problem (O’Neill, 1982), find out the claims solution that performs better than the others by reducing inequality promoting convergence to a greater degree (the constrained equal losses rule). Nonetheless, when using this egalitarian division of losses, regions may receive no amount of funds. This paper defines a new way to distribute the limited resources of the European Regional Development Fund (ERDF). We propose a compromise between the egalitarian approaches, i.e., we combine the egalitarian division of the funds with an egalitarian division of the losses (what regions do not get). In doing so, our proposal combines the constrained equal losses solution with the ensuring of a minimum amount to each region (sustainable bound). Finally, we provide an axiomatic analysis of the new solution and we apply it to the ERDF problem. Keywords: European Regional Development Fund; Conflicting claims problems; Egalitarian distribution; Constrained equal losses