Depleted Monolithic Active Pixel Sensor (DMAPS) prototypes developed in the TowerJazz 180 nm CMOS imaging process have been designed in the context of the ATLAS upgrade Phase-II at the HL-LHC. The pixel sensors are characterized by a small collection electrode (3 $mu$m) to minimize capacitance, a small pixel size ($36.4times 36.4$ $mu$m), and are produced on high resistivity epitaxial p-type silicon. The design targets a radiation hardness of $1times10^{15}$ 1 MeV n$_{eq}$/cm$^{2}$, compatible with the outermost layer of the ATLAS ITK Pixel detector. This paper presents the results from characterization in particle beam tests of the Mini-MALTA prototype that implements a mask change or an additional implant to address the inefficiencies on the pixel edges. Results show full efficiency after a dose of $1times10^{15}$ 1 MeV n$_{eq}$/cm$^{2}$.