Mocanu's Jacobi-form isomorphism conjecture for the lattices $D_3$, $D_5$, and $D_7$

Let $J_{k,D_r}$ denote the space of Jacobi forms of weight $k$ and lattice index $D_r$. Let $M_k^{new,\u03b5}(N)$ be the subspace of new elliptic modular forms of weight $k$ for $\Gamma_0(N)$ with Fricke eigenvalue $\epsilon i^{-k}$, and let $M_k^{}(N)$ denote the corresponding Fricke eigenspaces. Two such spaces are Hecke equivalent when the displayed isomorphism is compatible with their Hecke-module structures.

Mocanu's conjecture. For every $k \geq 2$, there are Hecke-equivalent isomorphisms

$$J_{k+2,D_3} \cong M_{2k}^{new,-}(2) \oplus M_{2k}^{+}(1),$$ $$J_{k+3,D_5} \cong M_{2k}^{new,+}(2) \oplus M_{2k}^{-}(1),$$ $$J_{k+4,D_7} \cong M_{2k}^{new,+}(2) \oplus M_{2k}^{+}(1).$$

The conjecture generalizes the known isomorphism for index $D_1$ (equivalently, index $2$), and is based on numerical examples of Euler factors of Jacobi forms. Its resolution is not established by the supplied text.