Very likely, you can still use an actual click to do it. If the click does not result in any unwanted filters applied or the drop-down thing closing, you can send those clicks, which will make each menu actually open.
Alternatively you could use Send keys as suggested by @catto16 or a JavaScript.
Finally, does the filter actually appear in the URL after you apply it? Perhaps it's possible to simply pass it to the URL instead of doing it via the UI.
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