  We complete the problem of finding the universal central extension in the
category of Leibniz superalgebras of $\mathfrak{sl}(m, n, D)$ when $m+n \geq 3$
and $D$ is a superdialgebra, solving in particular the problem when $D$ is an
associative algebra, superalgebra or dialgebra. To accomplish this task we use
a different method than the standard studied in the literature. We introduce
and use the non-abelian tensor square of Leibniz superalgebras and its
relations with the universal central extension.
