  Gut and Stadm\"{u}ller (2021, 2022) initiated the study of the elephant
random walk with limited memory. Aguech and El Machkouri (2024) published a
paper in which they discuss an extension of results by Gut and Stadtm\"{u}ller
(2022) for an "increasing memory" version of the elephant random walk without
stops. Here we present a formal definition of the process which has been hinted
at Eq. (2.2) in Gut and Stadtm\"{u}ller (2022). This definition is based on the
triangular array setting. We give a positive answer to the open problem in Gut
and Stadtm\"{u}ller (2022) for the elephant random walk, possibly with stops.
We also obtain the CLT for the supercritical case of this model.
