Holographic renormalization conjecture for bulk Laplacian eigenvalue variation

Let $(X,G)$ be the bulk with boundary $M$, and let $\tilde{D}=-\Delta_G$ and $D=-\Delta_g$ be the bulk and boundary Laplace--Beltrami operators, respectively. Here $\lambda$ is an eigenvalue of $\tilde{D}$, $\rho$ is the holographic renormalization parameter, and $\zeta(s,D)$ is the spectral zeta function of $D$. Holographic eigenvalue-variation conjecture. In the geometric setup of the AdS/CFT correspondence, holographic renormalization gives

$$\frac{\partial \lambda}{\partial \rho}=-\frac{1}{2}\zeta'(0,D)+\rho\zeta(0,D).$$

In the first-term approximation, this becomes

$$\frac{\partial \lambda}{\partial \rho}=-\frac{1}{2}\zeta'(0,D).$$

The claim connects spectral data of the boundary Laplacian with the variation of bulk eigenvalues and is motivated by the relation between thermal entropy, free energy, and holographic renormalization in odd-dimensional bulk geometry. The supplied text gives no evidence that the conjecture has been resolved.