Brumer–Kramer paramodularity conjecture for abelian surfaces

Let $B$ be an abelian surface defined over $\mathbb{Q}$ of conductor $N$ such that $\operatorname{End}_{\mathbb{Q}}(B)=\mathbb{Z}$. A Brumer–Kramer paramodularity conjecture. There exists a Siegel newform $g$ of genus $2$, weight $2$ and paramodular level $N$ such that

$$L(B,s)=L(g,s).$$

Conversely, if $g$ is a Siegel newform of genus $2$, weight $2$ and paramodular level $N$, which is a non-Gritsenko lift and whose Hecke eigenvalues are integers, then there exists an abelian surface $B$ defined over $\mathbb{Q}$ such that $\operatorname{End}_{\mathbb{Q}}(B)=\mathbb{Z}$ and

$$L(g,s)=L(B,s).$$

The conjecture predicts a correspondence between suitable abelian surfaces over $\mathbb{Q}$ and Siegel newforms of genus $2$ with matching $L$-functions. The paper exhibits an example satisfying the conjecture, but the source notes that none of the relevant surfaces has been proved to be modular, so the general statement remains unresolved.