Böcherer–Kikuta's generalization conjecture for mod $p^m$ singular forms

Let $n$, $k$, and $N$ be positive integers, let $r$ be an even integer, and let $p$ be a prime. Let $\chi$ be a quadratic Dirichlet character modulo $N$ satisfying $\chi(-1)=(-1)^k$. Suppose that $F\in M_k(\Gamma_0^{(n+r)}(N),\chi)_{\mathbb Z_{(p)}}$ is mod $p^m$ singular of $p$-rank $r$. The proven theorem assumes $n\ge r$ and $p>r+1$, and expresses $F$ modulo $p^m$ as a finite linear combination of theta series satisfying the stated modularity and level-divisibility conditions. Böcherer–Kikuta's generalization conjecture. Theorem general should hold for any prime, not only for $p>r+1$, and without the assumption $n\ge r$. This would extend the structure theorem to small primes and to the range $n<r$, where the theorem in the paper does not apply.