Boundary Polyakov–Alvarez asymptotic formula for Brownian loop measure

Let $(M,g)$ be a fixed compact smooth two-dimensional Riemannian manifold with smooth boundary, and let $\mu^{\text{loop}}$ denote the Brownian loop measure on $(M,g)$. Let $K$ be the Gaussian curvature on $M$, let $\Delta$ be the Laplacian associated to $(M,g)$, and let $\det_\zeta' \Delta$ denote its zeta-regularized determinant. Let $\mathcal B$ be a family of Lipschitz functions with uniformly bounded Lipschitz constants that is precompact in $W^{1,1}(M)$. For a conformal factor $\rho$, write $\operatorname{Vol}_\rho(M)$ and $\operatorname{Len}_\rho(\partial M)$ for the corresponding weighted volume and boundary length. Boundary Polyakov–Alvarez conjecture. The $\mu^{\text{loop}}$-mass of loops with $\rho$-length greater than $\delta$ should be

$$\frac{\operatorname{Vol}_{\rho}(M)}{2\pi\delta} - \frac{\operatorname{Len}_{\rho}(\partial M)}{2\sqrt{2\pi \delta}} - \frac{\chi(M)}{6} \left(\log \frac{\delta}{2} + \gamma\right) + \frac{1}{48\pi} \int_M (\| \nabla \rho \|^2 + 2 K \rho)\,\operatorname{Vol}(dz) + \log \operatorname{Vol}(M) - \log \operatorname{Vol}_\rho(M) - \log \det\nolimits_{\zeta}' \Delta + O(\delta^{1/2}),$$

with convergence as $\delta\to0$ uniform over $\rho\in\mathcal B$, where $\gamma\approx0.5772$ is the Euler–Mascheroni constant. This would extend the corresponding Brownian-loop asymptotic formula from closed surfaces to manifolds with boundary, including the boundary contribution of order $\delta^{-1/2}$ and the Polyakov–Alvarez-type curvature and conformal terms.