How do I launch an incognito search with the "Web Searches" plugin?

https://preview.redd.it/87t2zqynswoh1.png?width=259&format=png&auto=webp&s=a5206dcccd6c413fb95fcc0f761d2e1fbb765974

https://preview.redd.it/n1mvx1c4twoh1.png?width=814&format=png&auto=webp&s=1a0bd042402a910647a024090ebdbc9da02a4f79

In the Web Searches plugin, I have the private mode checkbox enabled the desired website, but when I type in a search query, it opens on a regular tab rather than an incognito tab.

If there is a solution can it also be applied to LinkOpener?

https://preview.redd.it/wutkpk1mtwoh1.png?width=1095&format=png&auto=webp&s=6feae694b42b3153476a2efe7f3b7a1975712fc7