Finite double coset index is not finite-intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., subgroup of finite double coset index) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).
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Statement

An intersection of finitely many subgroups, each having finite double coset index in the whole group, need not have finite double coset index in the whole group.

More specifically, it is possible to have two subgroups $H,K\le G$, both with finite double coset index in $G$, such that $H\cap K$ does not have finite double coset index in $G$.