Arlandini–Loeffler twisted factorization conjecture for adjoint $p$-adic $L$-functions

Let $\psi$ be a finite order Dirichlet character, let $U$ be the weight domain of a Coleman family $\mathcal{F}$, let $\mathcal{W}$ be the cyclotomic weight space, and let $\mathcal{F}_\psi=\mathcal{F}\otimes\psi$ denote its twist. Write $\mathbf{V}_\mathcal{F}$ for the Galois representation attached to $\mathcal{F}$, $\varepsilon_f$ for the nebentype character of the specialization $f$, and $L_p^{\rm geom}(\mathcal{F},\mathcal{F}_\psi)$ for the geometric Rankin $p$-adic $L$-function. The imprimitive adjoint $p$-adic $L$-function is denoted by $L_p^{\rm imp}(\operatorname{Sym}^2(\mathbf{V}_\mathcal{F}),\psi)$. Arlandini–Loeffler's twisted factorization conjecture. For every finite order Dirichlet character $\psi$, there exists a unique two-variable meromorphic $p$-adic $L$-function $L_p^{\rm imp}(\operatorname{Sym}^2(\mathbf{V}_\mathcal{F}),\psi)$ interpolating the critical values of $L^{\rm imp}(\operatorname{Sym}^2(f),\psi,s)$, and for all $(\sigma_1,\sigma_2)\in U\times\mathcal{W}$ one has

$$L_p^{\rm geom}(\mathcal{F},\mathcal{F}_\psi)(\sigma_1,\sigma_1,\sigma_2)=L_p^{\rm imp}(\operatorname{Sym}^2(\mathbf{V}_\mathcal{F}),\psi)(\sigma_1,\sigma_2)L_p(\varepsilon_f\psi,\sigma_2-\sigma_1-1).$$

This is the twisted analogue of the interpolation and factorization results of Arlandini and Loeffler, which are not established in the source for nontrivial characters $\psi$. The conjecture is assumed as an input for the paper's subsequent results.