  In this paper, we are interested in analyzing the dynamics of the
fourth-order difference equation $x_{n+4} =
\max\{x_{n+3},x_{n+2},x_{n+1},0\}-x_n$, with arbitrary real initial conditions.
We fully determine the accumulation point sets of the non-periodic solutions
that, in fact, are configured as proper compact intervals of the real line.
This study complements the previous knowledge of the dynamics of the difference
equation already achieved in [M. Cs\"ornyei, M. Laczkovich, Monatsh. Math. 132
(2001), 215-236] and [A. Linero Bas, D. Nieves Rold\'an, J. Difference Equ.
Appl. 27 (2021), no. 11, 1608-1645].
