Beltrametti–Sommese equidimensionality conjecture for adjunction-theoretic scrolls

Let $(X,L)$ be a polarised projective manifold of dimension $n$, and let $\varphi\colon X\rightarrow Y$ be an adjunction-theoretic scroll over a normal variety $Y$ of dimension $m$. Beltrametti–Sommese's conjecture. If $n\geq 2m-1$, then $\varphi$ is equidimensional. The conjecture is known when $L$ is very ample and $n\geq 2m$, with partial results for low-dimensional $Y$, but remains open in the stated generality.