Maximal TC weight from positive homological norm

Let $M$ be a closed, connected, oriented aspherical $n$-manifold and let $2 \le k \le n$; all homology and cohomology have rational coefficients. The $\ell^1$-seminorm of $\alpha \in H_k(M;\mathbb{Q})$ is $$\|\alpha\|_1 = \inf\Bigl\{\sum_j |a_j| : \sum_j a_j \sigma_j \text{ is a singular cycle representing } \alpha\Bigr\}.$$ For $u \in H^k(M;\mathbb{Q})$ put $\bar u = 1 \times u - u \times 1 \in H^k(M \times M;\mathbb{Q})$. Let $e : C([0,1], M) \to M \times M$, $\gamma \mapsto (\gamma(0), \gamma(1))$, be the endpoint fibration, the path space carrying the compact-open topology, and let $\operatorname{genus}$ denote the unreduced Schwarz genus: the least number of open sets covering the base on each of which the fibration has a continuous section. The TC weight of $\bar u$ is $$\operatorname{wgt}_{\mathrm{TC}}(\bar u) = \sup\bigl\{r \in \mathbb{Z}_{\ge 0} : f^*\bar u = 0 \text{ for every space } Y \text{ and every continuous } f : Y \to M \times M \text{ with } \operatorname{genus}(f^*e) \le r\bigr\}.$$ The genus of $e$ is the topological complexity of $M$. For a nonzero class the weight is at most its degree, so $\operatorname{wgt}_{\mathrm{TC}}(\bar u) \le k$ for $u \ne 0$; and the topological complexity exceeds the weight by at least one. Neofytidis proved $\operatorname{wgt}_{\mathrm{TC}}(\bar u) \ge 2$ for every nonzero $u$ of degree $k \ge 2$ on a closed oriented negatively curved manifold.

Problem. Suppose some $\alpha \in H_k(M;\mathbb{Q})$ has $\|\alpha\|_1 > 0$. Must there be a nonzero $u \in H^k(M;\mathbb{Q})$ with $\operatorname{wgt}_{\mathrm{TC}}(\bar u) \ge k$? The class $u$ may be chosen freely; it need not be dual to $\alpha$.