  We introduce $X$-to-join representations of inverse semigroups which are a
relaxation of the notion of a cover-to-join representation. We construct the
universal $X$-to-join Booleanization of an inverse semigroup $S$ as a weakly
meet-preserving quotient of the universal Booleanization ${\mathrm B}(S)$ and
show that all such quotients of ${\mathrm B}(S)$ arise via $X$-to-join
representaions. As an application, we provide groupoid models for the
intermediate boundary quotients of the $C^*$-algebra of a Zappa-Sz\'ep product
right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.
