The extension conjecture for projective-bundle divisors

Let $L$ be an ample line bundle on a projective manifold $X$ of dimension $n\ge 3$. Assume that there is a smooth divisor $Y\in |L|$ such that $Y$ is a $\mathbb{P}^{d}$-bundle, with morphism $p:Y\to Z$ over a manifold $Z$ of dimension $b$.

Extension conjecture. Then $d\ge b-1$, and $(X,L)\cong (\mathbb{P}(\mathcal{E}),\xi_{\mathbb{P}})$ for an ample vector bundle $\mathcal{E}$ on $Z$, with $p$ equal to the restriction to $Y$ of the induced projection $\mathbb{P}(\mathcal{E})\to Z$, except in one of the following cases:

1. $(X,L)\cong (\mathbb{P}^{3},\mathcal{O}_{\mathbb{P}^{3}}(2))$;
2. $(X,L)\cong (\mathcal{Q}^{3},\mathcal{O}_{\mathcal{Q}^{3}}(1))$;
3. $Y\cong \mathbb{P}^{1}\times\mathbb{P}^{n-2}$, $p$ is the product projection onto the second factor, and $(X,L)\cong (\mathbb{P}(\mathcal{E}),\xi_{\mathbb{P}})$ for an ample vector bundle $\mathcal{E}$ on $\mathbb{P}^{1}$, with the product projection of $Y$ onto the first factor equal to the induced projection $\mathbb{P}(\mathcal{E})\to\mathbb{P}^{1}$.

This conjecture describes the known examples in the extension problem for projective-bundle structures on ample divisors. The source gives no resolution status, so its general validity remains open.