Inverse flower arrangement of subgroups

Definition
Let $$G$$ be a group and $$H_i, i \in I$$, be a collection of subgroups of $$G$$. The $$H_i$$ form an inverse flower arrangement of subgroups if the join of any two $$H_i$$s is the same: for any $$i \ne j \in I$$ and any $$k \ne l \in I$$, we have:

$$\langle H_i, H_j \rangle = \langle H_k, H_l \rangle$$.

(Note that $$i,j$$ are allowed to equal $$k,l$$).

This is closely related to the notion of flower arrangement of subgroups and flower arrangement of subsets.