Universality conjecture for split localization entropy on a halfline

The transverse symmetry and absence of correlations reduce the area-density problem to a chiral theory on a lightray, which is mapped by a Cayley transform to a chiral theory on the circle. Let $I$ and $I^{\prime}$ be opposite half-circle intervals, and let $I_{\varepsilon}$ and $I_{\varepsilon}^{\prime}$ separate their ends by a distance $\varepsilon$. For a chiral split inclusion

$$\mathcal{A}(I_{\varepsilon})\subset\mathcal{N}_{\varepsilon}\subset\mathcal{A}(I),\qquad \mathcal{B}(\mathcal{H})=\mathcal{N}_{\varepsilon}\otimes\mathcal{N}_{\varepsilon}^{\prime},$$

write $S_{\varepsilon}$ for the split localization entropy of the halfline, with the sharp localization limit corresponding to $\varepsilon\rightarrow0$. Universality conjecture. The split localization entropy behaves as

$$S_{\varepsilon}\underset{\varepsilon\rightarrow0}{=}-c\ln\varepsilon,$$

where $c$ depends on the chiral model appearing in the holographic projection, namely its holographic universality class. Consequently, the horizon-associated entropy area density is determined by these considerations only up to an overall normalization independent of the holographic universality class. This conjecture proposes universal logarithmic divergence of the split entropy; the source states that a proof was not available, and no resolution is supplied here.