Meromorphic theta-function conjecture for generalized $H^{q,k}$

Let $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$. Let $H^{q,k}(z,v,\omega,\lambda)$ be the integral defined by

$$H^{q,k}(z,v,\omega,\lambda):=\frac{1}{2\pi i}\oint_{I(\lambda,\infty)}e^{(\lvert\omega\rvert-z)t}J^k(\mathfrak{g}(u))(1)\frac{\log(-t)^q}{t}\,dt,$$

where $J$ and $\mathfrak{g}$ are as defined in the preceding discussion, and set $L=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$. Meromorphic theta-function conjecture. For all $k,q\in\mathbb{N}$, $v\in\mathbb{C}$, and $\omega,\lambda\in\mathbb{R}_{>0}^2$, the integral defining $H^{q,k}(z,v,\omega,\lambda)$ defines a meromorphic function, which as a function of $z$ is a theta function for $L$.

The preceding proposition establishes convergence and analyticity only in a right half-plane under weaker hypotheses. This conjecture proposes meromorphic continuation in $z$ for positive real parameter vectors while retaining the theta-function property.