Ibukiyama's transfer conjecture for paramodular Siegel modular forms

Let $D$) be the quaternion algebra over $\mathbb{Q}$ that is non-split over $\mathbb{R}$ and $\mathbb{Q}_p$ for a prime $p$ and split at all other places. Let $G=GU_2(D)$ be the associated quaternionic similitude group, and let $S_{k,j}[K(p)]^{\text{new}}$ be the space of $p$-new Siegel modular forms of weight $k,j$ and paramodular level $K(p)$. Let $\mathcal{A}^G_{k,j}[K_2(p)]$ be the corresponding space of algebraic modular forms for $G$ of level $K_2(p)$. Ibukiyama's conjecture. For $k \ge 0$ and $j \ge 3$ there is an injective map

$$S_{k,j}[K(p)]^{\text{new}} \xhookrightarrow{} \mathcal{A}^G_{k,j}[K_2(p)],$$

which is Hecke-equivariant for the prime-to-$p$ Hecke operators. This is a conjectural Jacquet–Langlands-type transfer from the quaternionic similitude group to paramodular Siegel modular forms; the supplied text gives no resolution status.