Duke–Li valuation conjecture for CM coefficients

Let $l>3$ be a prime with $l\equiv 3\pmod{4}$, let ${\mathsf{k}}=\mathbb{Q}(\sqrt{-l})$, and let $p$ be a prime that is non-split in ${\mathsf{k}}$. Let $\mathfrak{P}_0\mid p$ be the unique prime above $p$ fixed by complex conjugation, and choose a fractional ideal $\mathfrak{b}$ such that $\mathfrak{P}^{\sigma_{\mathfrak{b}}}=\mathfrak{P}_0$. For the scalar-valued harmonic weak Maaß forms ${\widetilde{\theta}}_{\mathfrak{a}}(\tau)$, write $c^+_{\mathfrak{a}^2}(n)$ for the relevant coefficient and let $\rho(m,\mathcal{C})$ denote the number of integral ideals of $\mathcal{O}_{\mathsf{k}}$ of norm $m$ in the ideal class $\mathcal{C}$. Duke–Li's valuation conjecture. The functions ${\widetilde{\theta}}_{\mathfrak{a}}(\tau)$ can be chosen so that, for $n\in\mathbb{Z}_{>0}$ with $\chi_p(n)\ne1$,

$$c^+_{\mathfrak{a}^2}(n)=-\frac{2}{r}\log\left\lvert u(\mathfrak{a}^2,n)\right\rvert,$$

where $u(\mathfrak{a}^2,n)\in\mathcal{O}_{\mathsf{H}_{\mathsf{k}}}$ and, if $\operatorname{Diff}\left(\frac{n}{\lvert D\rvert}\right)=\{p\}$, then

$$\operatorname{ord}_{\mathfrak{P}}(u(\mathfrak{a}^2,n))=2r\sum_{m\ge1}\rho\left(\frac{n}{p^m},[\mathfrak{a}\mathfrak{b}]^2\right),$$

with $r\in\mathbb{Z}$ independent of $n$ and satisfying $r\mid24h_{\mathsf{k}}h_{\mathsf{H}_{\mathsf{k}}}$. The condition involving $\operatorname{Diff}$ is part of the conjecture, and the stated normalization accounts for the factor $2$ on the right-hand side. The source presents this as a conjecture motivated by numerical experiments; no resolution is supplied.