Prove the restricted quantum focusing conjecture for semiclassical gravity
In plain words
The classical singularity theorems assume energy is never negative, which quantum fields violate. A quantum replacement based on the entropy of the fields plus the area of a surface would restore the theorems, but its original strong form fails in simple models and a weaker form is still a conjecture.
Precise statement
Define the generalized entropy $S_{\mathrm{gen}} = A c^3/(4 G \hbar) + S_{\mathrm{out}}$ of a codimension-2 surface (entropy in units of $k_{\mathrm{B}}$, $S_{\mathrm{out}}$ the entropy of fields outside it) and the quantum expansion $\Theta$ as its variation per unit transverse area along a null congruence with affine parameter $\lambda$. The original quantum focusing conjecture dTheta/dlambda <= 0 (Bousso, Fisher, Leichenauer, Wall 2015) has explicit counterexamples in two-dimensional dilaton gravity coupled to quantum fields (2025); the restricted version requires dTheta/dlambda <= 0 only where $\Theta = 0$ (Shahbazi-Moghaddam 2022). Prove the restricted conjecture for interacting quantum fields coupled to semiclassical gravity in spacetime dimension $d > 2$, or construct a violation.
What would settle it
A proof of restricted quantum focusing in $d > 2$ for interacting fields, or an explicit violating configuration.
Status in the literature
Unverified note
Restricted focusing is proved in braneworld models (2022) and in JT gravity coupled to a field theory, where the original conjecture is also shown to fail (Franken, Kaya, Rondeau, Shahbazi-Moghaddam, Tran, arXiv:2510.13961, 2025); the general case in $d>2$ is open as of 2026.