Nonvanishing of special $L$-values of cusp forms on $\mathrm{GL}(2)$ with totally split prime power twists: without the class number hypothesis

Let $F$ be a number field and $f$ a normalized cuspidal Hecke eigenform on $\mathrm{GL}(2)$ over $F$, with $k/2$ as the central critical point of its $L$-function $L(f,s)$. Let $\mathfrak{p}$ be a totally split prime ideal of $F$ lying above an odd prime $p$ and coprime to the level ideal of $f$. In