Algebraic principal-part conjecture for harmonic weak Maass forms

Let $m>0$ and let $f$ be a harmonic weak Maass form of weight $\frac{1}{2}-m$ for the representation $\rho_L$. Assume that $\xi_{\frac{1}{2}-m}f$ is holomorphic also at $\infty$, and that the principal part of $f$ is algebraic, meaning that it has only Fourier coefficients algebraic over $\mathbb{Q}$, or even integral. The condition

$$\xi_{\frac{1}{2}-m}f\neq 0$$

then implies that $f$ is not weakly holomorphic and that some Fourier coefficient of the holomorphic part of $f$ is transcendental over $\mathbb{Q}$.

This predicts that a non-weakly-holomorphic harmonic weak Maass form with algebraic principal part cannot have all coefficients of its holomorphic part algebraic. The assertion is used to distinguish algebraic theta lifts from those arising from harmonic weak Maass forms in the case $b_{-}=1$; its resolution is not specified in the supplied text.