Miller's exact geometric conjecture for the heat observability constant

Let $({\mathcal M},g)$ be a Riemannian manifold and let $\omega\subset {\mathcal M}$ satisfy $\overline{\omega}\neq {\mathcal M}$. Define

$${\mathcal L}({\mathcal M},\omega)=\sup_{x\in {\mathcal M}}\operatorname{dist}_g(x,\omega).$$

Let ${\mathfrak K}_{heat}(\omega)$ denote the short-time heat observability constant. Miller's conjecture.

$${\mathfrak K}_{heat}(\omega)=\frac{{\mathcal L}({\mathcal M},\omega)^2}{4}.$$

This conjecture makes precise the proposed relationship between heat observability and the maximal distance from the observation set. Miller proved the lower bound ${\mathfrak K}_{heat}(\omega)\geq {\mathcal L}({\mathcal M},\omega)^2/4$, while the equality for all such manifolds and observation sets remains open in the source.