Fixed-denominator joint Gaussian distribution conjecture for additive twists

Let $(\phi_j)_{j=1}^r$ be a finite orthogonal family of Hecke–Maaß cusp forms for $\operatorname{SL}(2,\mathbb{Z})$. Let $\mathcal{V}(x)$ be the associated vector of additive-twist values, let $\Sigma$ be the covariance matrix from the joint central-limit theorem, and let $R\subset\mathbb{R}^{2r}$ be a measurable set with boundary of measure zero. Fixed-denominator joint Gaussian conjecture. The normalized values at reduced fractions satisfy

$$\mathbb{P}_q\left(\frac{\mathcal{V}(a/q)}{\sqrt{\log q}}\in R\right)\longrightarrow \mathbb{P}\left(\mathcal{N}(0,\Sigma)\in R\right)\qquad(q\to\infty),$$

where the probability is taken over $a\in(\mathbb{Z}/q\mathbb{Z})^\times$. The paper proves the corresponding result after averaging over denominators and conjectures that this extra average over $q$ is unnecessary.