Andreatta–Wiśniewski equality conjecture for extremal fibers

Let $\varphi:X\rightarrow Z$ be a Fano–Mori contraction of an extremal ray $R$ of a smooth variety $X$. Let $E$ be its exceptional locus, with $E=X$ if $\varphi$ is of fiber type, let $S$ be an irreducible component of a non-trivial fiber, and set

$$l=\min\{-K_X\cdot C:C\text{ is a rational curve in }S\}.$$

Andreatta–Wiśniewski's equality conjecture. If

$$\dim S+\dim E=\dim X+l-1,$$

then the normalization of $S$ is isomorphic to $\mathbb P^s$. The statement is posed as a conjecture after a proposition proving the corresponding conclusion in the supported-contraction setting; the source gives no resolution status.