Volume version of Penrose's singularity theorem

Let $(M,g)$ be a globally hyperbolic spacetime satisfying the strong energy condition, and let $S\subset M$ be a future trapped surface. Volume version of Penrose's singularity theorem. Then

$$\operatorname{vol}_g\left(I^+(S)\right)<\infty.$$

This is proposed as a volume analogue of Penrose's singularity theorem: whereas Penrose's theorem gives geodesic singularities under generic conditions, this conjecture would imply formation of an event horizon. The assumptions are described as tentative, and the stronger energy condition is used because it gives stronger control over volumes.