Algebraic Fourier coefficients of harmonic Maass forms

Let $\tilde f$ be a harmonic Maass form of weight $k \in \frac{1}{2}\mathbb{Z}$ for a congruence subgroup, and suppose that $\xi(\tilde f) \neq 0$. Its holomorphic part is the holomorphic part of the Fourier expansion of $\tilde f$.

Algebraicity conjecture. If all Fourier coefficients of the holomorphic part of $\tilde f$ are algebraic, then $\xi(\tilde f)$ is either a unary theta series with $k \in \{\frac{1}{2},\frac{3}{2}\}$ or a binary theta series with $2-k \geq 2$. In these cases, the coefficients have bounded denominators if and only if $k$ is half-integral.

The conjecture formalizes the expectation that the known examples with algebraic Fourier coefficients are the only ones. The source notes that generic harmonic Maass forms are generally believed to have transcendental Fourier coefficients, while the unary case has rational coefficients with explicitly bounded denominators and the binary case appears to have algebraic coefficients with unbounded denominators. The conjecture's general status is not specified in the source.