Spectral fourth moment conjecture for weight 1 Maass forms

Let $N$ be the level, and let $\{\mu_j^1\}_j$ be an orthonormal basis of Maass forms of weight $1$ on $\Gamma_0(N)$, with spectral parameters $t_j$ and Fourier coefficients $\rho_j^1(1)$. Fix $r\in\mathbb{R}$ and $\epsilon>0$. Spectral fourth moment conjecture for weight $1$ Maass forms. As $T\to\infty$, one has

$$\sum_{\lvert t_j\rvert\leq T}\frac{\lvert\rho_j^1(1)\rvert^2}{\cosh(\pi t_j)}\left\lvert L\left(\tfrac12+ir,\mu_j^1\right)\right\rvert^4\ll_{N,r,\epsilon}T^{1+\epsilon}.$$

This conjectural fourth-moment estimate would extend the well-understood spectral fourth-moment bounds for weight $0$ Maass forms to weight $1$ and, through raising and lowering operators, to other odd weights. Such estimates are intended to control the discrete-spectrum contribution in shifted-convolution Dirichlet series and are presented as unavailable in the existing literature.