  A Banach space is locally almost square if, for every $y$ in its unit sphere,
there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm
x_n\|=1$. A Banach space is weakly almost square if, in addition, we require
the sequence $(x_n)$ to be weakly null. It is known that these two properties
are distinct, so we aim to investigate if local almost squareness implies a
weaker version of the latter property by replacing the sequence with a net. In
order to achieve this result, we restrict ourselves to Banach lattices and
introduce a strengthening of local almost squareness by requiring that the
sequence is in the positive cone of the lattice. As an application of such
characterization, we prove that this positive variant of local almost
squareness implies that every relatively weakly open set in the unit ball has
diameter 2, that is, the Banach space has the so called diameter two property.
This in particular allows us also to generate new examples of Banach spaces
enjoying the diameter two property.
