The linear Green function sphere-sum conjecture for finitely generated nilpotent groups

Let $\Gamma$ be a finitely generated nilpotent group, let $\mu$ be a symmetric admissible finitely supported probability measure, and let $S$ be a finite generating set. Writing $S_n$ for the sphere of radius $n$ in the word metric associated with $S$, and $G(e,x)$ for the Green function of the random walk driven by $\mu$, the linear Green function sphere-sum conjecture.

$$\sum_{x\in S_n}G(e,x)\asymp n.$$

This is presented as a reformulation of the conjecture of Breuillard and Le Donne that the exponent $\beta$ in the error estimate for nilpotent-group ball growth can always be taken to be $1$. The statement is intended to hold for all finitely generated nilpotent groups and the specified measures and generating sets; its status is not resolved in the supplied context.