Debarre–Jiang–Voisin strong and weak conjectures for contracted cycles

Let $\pi:X\to Y$ be a morphism of projective varieties over an algebraically closed field. Let $N_k(X)$ be the space of real $k$-cycle classes modulo numerical equivalence, and let $\operatorname{\overline{Eff}}_k(X)$ be the closure of the cone generated by effective $k$-cycles. Suppose that $\alpha\in\operatorname{\overline{Eff}}_k(X)$ satisfies

$$\pi_*\alpha=0.$$

Debarre–Jiang–Voisin conjecture. The class $\alpha$ lies in the vector space generated by $k$-dimensional subvarieties contracted by $\pi$ (Weak Conjecture), and in fact lies in the closure of the cone generated by such subvarieties (Strong Conjecture). This predicts that pseudo-effective classes in the kernel of pushforward are built from cycles contracted by the morphism; the paper establishes new cases, including a strong version for morphisms from fourfolds of relative dimension one, while the general statement remains open.