A positive-norm counterexample to a TC-weight question

Let $S$ be the closed oriented surface of genus two. The aspherical four-manifold $M=S\times S$ has a rational fundamental class of positive homological $\ell^1$ seminorm, but every nonzero $u\in H^4(M;\Q)$ satisfies $\wt(1\times u-u\times1)\le2$. This gives a negative answer to Neofytidis's question asking whether a positive-norm degree-$k$ class on a closed oriented aspherical manifold forces the existence of a nonzero degree-$k$ class with TC weight at least $k$.