A Complete Proof of Mueller--Ho Conjecture

For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $θ(s;z)=\sum_{m,n\in\mathbb Z}e^{-πs\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-π\big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2π(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice