Fourier-coefficient relation conjecture for Jacobi cusp forms

Let $k>4$ with $k\equiv 2$ modulo $4$. For every reduced matrix $T\in\halfint^+_2$ with bottom-right entry $m$, let $S_m(n,T)$ be the set of integers $r$ such that $\widetilde{T}=\smT\in\halfint^+_2$ and, for $j=1,\ldots,d$, the displayed ratios of Fourier coefficients satisfy

$$\frac{a^k_2(\Tdiv{\widetilde{T}}{t_j})}{a^k_2(\widetilde{T})}=\frac{a^k_2(\Tdiv{T}{t_j})}{a^k_2(T)}.$$

Here $c_{n,r}$ denotes the corresponding functional on $J_{k,m}^{\mathrm{cusp}}(\mathbb{Q})$. Fourier-coefficient relation conjecture. For every such $T$, there exist arbitrarily large integers $A$ and positive rational numbers $\mu_{n,r}$ such that

$$\sum_{1\leq n\leq A}\sum_{r\in S_m(n,T)}\mu_{n,r}c_{n,r}\big|_{J_{k,m}^{\mathrm{cusp}}(\mathbb{Q})}=0.$$

This conjecture is presented as a refinement of an earlier lemma and is intended to imply the existence of the subcones needed for the generalization to every non-squarefree $m$; the supplied excerpt does not establish it.