The group-ring resolution conjecture for symmetric spaces

Let $\widetilde M^d$ be a symmetric space with curvature $-1\leq K\leq 0$ and real rank $r$. Let $G$ act on $\widetilde M$, and suppose there is a minimum displacement greater than $f(n)$, meaning

$$d(gx,x)>f(n)$$

for every $x\in\widetilde M$ and every $1\neq g\in G$.

Group-ring resolution conjecture. There is a function $f(n)$ such that every $n$-generated ideal in $\mathbb K[G]$ has a free resolution of length $r-1$.

Such a resolution property would imply, via the methods of the paper, $k$-cellular waist inequalities for maps from uniform locally symmetric spaces of real rank $r$ whenever $0<k<d-m-(r-1)$. The conjecture extends the hyperbolic-space case, where the corresponding group-ring ideals are free.