SO(5) mass-formula congruence conjecture for Siegel modular forms

Let $G$ be the relevant inner form of $\operatorname{SO}(5)$, let $K$ be the maximal compact subgroup in the notation of the SO(5) mass formula, and let $S=S_1\sqcup S_2\sqcup S_3$ be its finite ramification set with the indicated local types. Write $m(K)$ for the associated mass, let $\mathfrak p_v$ denote the prime at $v$, and let $\rho_3$ be the specified archimedean type. SO(5) mass-formula congruence conjecture. If $p\mid m(K)$, then there exists a holomorphic or non-holomorphic degree-$2$ Siegel modular form $f$ of archimedean type $\rho_3$ such that: (i) $f$ is unramified outside $S$; (ii) $f$ has Klingen level $\mathfrak p_v$ for $v\in S_1$; (iii) $f$ has paramodular level $\mathfrak p_v$ for $v\in S_2\cup S_3$; and (iv) $f$ is Hecke congruent modulo $p$ to the trivial representation of $\operatorname{PGSp}(4)$. The conjecture predicts that divisibility of the SO(5) mass produces congruences with the precise local level structure prescribed by the genus. The paper explains that the paramodular case is known, while the general behavior of level structure under the conjectural Jacquet--Langlands correspondence, and hence the full congruence statement, remains open.