Sharp timelike asymptotic volume ratio inequality for singularity formation

Let $(X,\mathsf d,\mathfrak m, \ll, \leq, \tau)$ be a measured Lorentzian space, let $V\subset X$ be the achronal set occurring in the hypotheses of Theorem, and let $\mathrm{TAVR}_{V,N}(X)$ denote its $N$-timelike asymptotic volume ratio. Write $\mathfrak m^+(V)$ for the future boundary measure and $I^-(V)$ for the chronological past of $V$. Sharp timelike asymptotic volume ratio conjecture. For every $N\ge 2$ there exists a constant $C_N>0$ such that, if $X$ and $V\subset X$ are as in Theorem, then

$$C_N\;\mathrm{TAVR}_{V,N}(X)\;\mathfrak m(I^-(V))^{N-1} \le \mathfrak m^+(V)^N.$$

This is proposed as a sharp form of the preceding singularity-volume estimate, with validity in every dimension. The conjecture relates the timelike asymptotic volume growth of the future of $V$ to the volume of its chronological past and its future boundary measure; its resolution is not supplied in the source.