  A recent result of T.~Abrahamsen, P.~H\'ajek and S.~Troyanski states that a
separable Banach space is almost square if and only if there exists $h\in
S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in X$. The proof
passes through a sequential version of being almost square which we call being
\textit{sequentially almost square}. In this article we study these conditions
in the nonseparable setting. On one hand, we show that a Banach space $X$
contains a copy of $c_0$ if and only if there exists an equivalent renorming $|
\cdot |$ on $X$ for which there exists $h\in S_{X^{****}}$ such that
$|x+h|=\max\{|x|,1\}$ for every $x\in X$. On the other hand, although it is
unclear whether the aforementioned result of T.~Abrahamsen et al. holds in the
nonseparable setting, we show that, under the existence of selective
ultrafilters, if $X$ is a sequentially almost square Banach space then there
exists $h\in S_{X^{****}}$ such that $\|x+h\|=\max\{\|x\|,1\}$ for all $x\in
X$.
