Sommese's classification conjecture for manifolds containing an ample projective-space bundle

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth ample divisor. Suppose that $p:Y\to Z$ exhibits $Y$ as a $\mathbb{P}^d$-bundle over a $b$-dimensional smooth variety $Z$. Sommese's conjecture. One of the following holds:

1. $X\simeq\mathbb{P}^3$, $Y\simeq\mathbb{P}^1\times\mathbb{P}^1$ is a smooth quadric, and $p$ is one of the projections to $\mathbb{P}^1$.
2. $X\simeq Q^3\subset\mathbb{P}^4$ is a smooth quadric threefold, $Y\simeq\mathbb{P}^1\times\mathbb{P}^1$ is a hyperplane section, and $p$ is a projection to one of the factors.
3. $Y\simeq\mathbb{P}^1\times\mathbb{P}^b$, $Z\simeq\mathbb{P}^b$, $p:Y\to Z$ is the projection to the second factor, and $X$ is the projectivization of an ample vector bundle $\mathscr{E}$ on $\mathbb{P}^1$.
4. $X\simeq\mathbb{P}(\mathscr{E})$ for an ample vector bundle $\mathscr{E}$ on $Z$, and $\mathscr{O}_X(Y)\simeq\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1)$, so $Y$ is a fiberwise hyperplane.

This is a classification conjecture for smooth projective varieties containing an ample divisor that is a projective-space bundle. The case $d\geq 2$ is known, as are the cases $d=1$ and $b=1,2$; the paper proves the case $\rho(Z)=1$ and reduces the general case to a further conjecture.