Everywhere convergence of one-parameter Eisenstein series beyond the proved range

Let $E_s(g)$ be the Eisenstein series induced by the parameter $s$, and let $A'$ be the cone used in the paper. Let $K$ be the unitary form and $N$ the unipotent subgroup in the rank $2$ hyperbolic Kac--Moody group. One-parameter convergence conjecture. $E_s(g)$ converges absolutely for $g\in KA'N$ and $\mathrm{Re}~s<-1-\gamma^{-1}$. The theorem proves almost-everywhere convergence when $\mathrm{Re}~s<-2$. The conjecture is suggested by the asymptotic estimates for the constant term and would extend convergence to all of $N$ throughout the larger parameter range.