Everywhere convergence of Eisenstein series under a weakened Godement criterion

Let $G$ be a rank $2$ hyperbolic Kac--Moody group, let $K\backslash G_\mathbb{R}/G_\mathbb{Z}$ be the arithmetic quotient, and let $E_\nu(g)$ be the Eisenstein series induced from a quasi-character $\nu$ of a Borel subgroup. Write $A'$ for the cone

$$A'=\{a\in A^+:a^{\alpha_i}<1,\ i\in I\}.$$

Everywhere-convergence conjecture. $E_\nu(g)$ converges absolutely for $g\in KA'N$ and $\nu$ satisfying $\mathrm{Re}~\nu(h_{\alpha_i})<-1$, $i\in I$. The proved result gives almost-everywhere convergence under Godement's stronger criterion; the conjecture weakens that condition and asserts convergence for every element of $N$, rather than outside a measure-zero subset.