Hello, to make sure we are talking the same thing, "when closing search" means "when clicking on the cancel button" at right of the search box (the little cross). This is not closing BSP2 itself.
So the statement is on bookmark panel displayed inside BSP2, below the search results panel, and not on the FF native sidebar, which no add-on can modify nor interact with.

Now, there are options in BSP2 options page, disabled by default (you should verify in your case), to adjust behavior of staying open or not. See "After show bookmark ..." options in the Search results section, and modify to your taste.

Hope this is what you are looking for, else open an issue on https://github.com/aaFn/Bookmark-search-plus-2/issues.
Thank you, aaFn.