Ervedoza–Zuazua finite-time observability constant conjecture

Let $L>0$ and $T>0$, and let $C_{\int}(T,L)$ be a constant for which the finite-time weighted observability estimate

$$\int_0^\infty\int_0^L e^{-\frac{L^2}{2t}}|\varphi(t,x)|^2\,dx\,dt\leqslant C_{\int}(T,L)\int_0^T|\partial_x\varphi(t,0)|^2\,dt$$

holds for the relevant solutions of the one-dimensional heat equation. Ervedoza–Zuazua conjecture. For every $\delta>0$ and $L>0$, one can choose $C_{\int}(T,L)$ such that

$$C_{\int}(T,L)=O\!\left(e^{\frac{\delta}{T}}\right)\quad\text{as }T\to0.$$

This conjecture concerns the sharp growth, as the control time tends to zero, of the finite-time observability constant obtained from the corresponding infinite-time estimate. The source attributes the conjecture to work of Ervedoza and Zuazua and gives no resolution.