The quantum focusing conjecture and the improved energy condition

By rearranging its terms, the Quantum Focusing Conjecture (QFC) can be viewed as a quantum energy condition, and we can consider various limits. A recent restricted version is a limiting form where the quantum focusing vanishes Θ → 0, and has been proven for Braneworld scenario. As a result, we derive an improved quantum null energy condition (INEC) $T_{\mathit{kk}}\ge \frac{ħ}{2\pi \mathcal{A}}\left(S_{\text{out}}^{\prime \prime }-\frac{1}{2}\theta S_{\text{out}}^{\prime }\right)$ \( {T}_{kk}\ge \frac{\mathrm{\hbar}}{2\pi \mathcal{A}}\left({S}_{\textrm{out}}^{\prime \prime }-\frac{1}{2}\theta {S}_{\textrm{out}}^{\prime}\right) \) , that can be proven with field theory techniques. We sketch the beginning of a proof, and briefly discuss possible interpretations in the absence of one.