What is an explicit example a pure polarizable finite-dimensional $\mathbb{Q}$-Hodge structure that is not a subquotient of the cohomology of a scheme smooth proper over $\mathbb{C}$? What if replace "a scheme smooth proper over $\mathbb{C}$" with "closed complex manifold bimeromorphic to a closed Kaehler manifold"?

Note that we are asking for an explicit example, not "Hilbert scheme blah blah countable blah blah".

If it helps, assume any famous conjecture you want to assume.