Gromov's vanishing conjecture for functorial seminorms

Let a functorial seminorm assign to every topological space $X$ a seminorm on singular homology with real coefficients, such that for every continuous map $f\colon X\to Y$ and every homology class $\alpha$ one has

$$\|f_*(\alpha)\|\leq\|\alpha\|.$$

Gromov's conjecture. Every functorial seminorm on singular homology with real coefficients vanishes for simply connected spaces.

The conjecture generalizes the vanishing of the seminorm induced by the $\ell^1$-norm on simply connected spaces. The source gives no resolution status.