The section and universal-bundle obstruction conjecture for dual abelian fibrations

Let $X\to\mathbb P^n$ be an abelian fibration, let $P\to\mathbb P^n$ be its relative Picard, and let $X_0$ be the relative Picard of $P$. Let $\beta_X\in\mathbb H^2(P,\mathcal O_P^*)$ be the obstruction class to the existence of a global universal bundle on $X\times_{\mathbb P^n}P$. The section and universal-bundle obstruction conjecture. The following conditions are equivalent:

1. $X$ is isomorphic to $X_0$.
2. $X$ admits a section.
3. There exists a universal bundle over $X\times_{\mathbb P^n}P$.

The obstruction to all three existence problems is $\beta_X$. The source says the assertion is essentially proved away from singular fibres, with singular fibres remaining the concern.