Twisted cubic moment conjecture for $GL(3)\times GL(2)$ $L$-functions

Let $a\geq 1$ be an integer, let $\alpha_1,\alpha_2,\alpha_3$ be the shifts, let $H$ be the test function, and let $(\phi_j)_{j=1}^{\infty}$ be an orthogonal basis of even Hecke-normalized Maass cusp forms for $SL_2(\mathbb{Z})$ satisfying

$$\Delta\phi_j=\left(\frac14-\mu_j^2\right)\phi_j.$$

Write $\epsilon\cdot\alpha=(\epsilon_1\alpha_1,\epsilon_2\alpha_2,\epsilon_3\alpha_3)$ and define

$$\tau_{\alpha}(n)=\sum_{d_1d_2d_3=n}d_1^{-\alpha_1}d_2^{-\alpha_2}d_3^{-\alpha_3}.$$

Twisted cubic moment conjecture. The full set of main terms for

$$\sum_{j=1}^{\infty}H(\mu_j)\frac{\lambda_j(a)\prod_{i=1}^{3}\Lambda(1/2-\alpha_i,\phi_j)}{\langle\phi_j,\phi_j\rangle}+\frac{1}{4\pi}\int_{\mathbb{R}}H(i\mu)\frac{\sigma_{-2\mu}(a)a^{-\mu}\prod_{i=1}^{3}\Lambda(1/2+i\mu-\alpha_i)\Lambda(1/2+i\mu+\alpha_i)}{|\Lambda(1+2i\mu)|^2}\,d\mu$$

is

$$\frac12\sum_{\epsilon_1,\epsilon_2,\epsilon_3=\pm1}\prod_{1\leq i<k\leq3}\zeta(1-\epsilon_i\alpha_i-\epsilon_k\alpha_k)\,a^{-1/2}\prod_{p\mid a}\left\{\tau_{-\epsilon\cdot\alpha}\left(p^{o_p(a)}\right)-\frac{\tau_{-\epsilon\cdot\alpha}\left(p^{o_p(a)-1}\right)}{p}\right\}\int_{(0)}\frac{H(\mu)}{|\Gamma(\mu)|^2}\prod_{i=1}^{3}\prod_{\pm}\Gamma_{\mathbb{R}}\left(\frac12\pm\mu-\epsilon_i\alpha_i\right)\frac{d\mu}{2\pi i}.$$

The sign triple $(+1,+1,+1)$ is the $0$-swap term. The conjecture extends the untwisted cubic moment formula previously stated and proved by the authors; establishing the complete collection of main terms in the twisted setting is the remaining issue.