A semisimple subcategory of Khovanov's Heisenberg category

We show the existence of a semisimple replete subcategory of Khovanov’s Heisenberg category that retains the isomorphism data of objects for the full category. This leads to a noncommutative tensor-triangular geometric example of a monoidal triangulated category whose Balmer spectrum satisfies the tensor product property but which contains one-sided thick tensor-ideals that are not two-sided, and whose standard support varieties fail to classify one-sided thick tensor-ideals.