Lehmann's conjecture on the Iitaka dimension of contracted cycles

Let $X$ be a projective variety of dimension $n$, let $\pi:X\to Z$ be a surjective morphism of projective varieties of relative dimension $e$, and let $\alpha\in\operatorname{\overline{Eff}}_k(X)_{\mathbb{Z}}$ have $\pi$-contractibility index $c$, meaning that $c$ is the largest non-negative integer at most $k$ such that $\alpha\cdot\pi^*A^{k-c+1}=0$ for an ample divisor $A$ on $Z$. Lehmann's conjecture. If

$$k-\dim Z<c<e,$$

then

$$\kappa(\alpha)\leq n-c.$$

The surrounding statement records the other expected bounds as theorems, while this intermediate-contractibility case is presented as the conjectural part. It concerns how the Iitaka dimension is constrained when a cycle is contracted by a morphism without falling into the two extremal cases.