The higher-dimensional filling volume accumulation conjecture

Let $M$ be a finite-volume hyperbolic $n$-manifold with torus cusps, let $V$ be its complete hyperbolic interior, and let $M(T_1,\ldots,T_m)$ denote the manifold obtained by filling the cusps along totally geodesic codimension-$1$ tori $T_1,\ldots,T_m$. Higher-dimensional filling volume conjecture. The volumes

$$\left\|M(T_1,\ldots,T_m)\right\|$$

accumulate on

$$\frac{\operatorname{Vol}(V)}{v_n}$$

but do not attain it. This proposes that the strict volume decrease and limiting-volume behavior of hyperbolic Dehn filling in dimension three persist for these higher-dimensional fillings.