Ervedoza–Zuazua small-time heat observability cost conjecture

Let $L>0$ and let $C_D(T,L)$ denote the boundary control cost for the one-dimensional heat equation on an interval of length $L$. Define the small-time exponential rate $\beta^*$ by

$$\beta^*:=(L^2/4)^+.$$

Ervedoza–Zuazua conjecture. For every $L>0$ and every $K>L^2/4$, there exists some $C(K)>0$ such that, for every $T$ sufficiently small,

$$C_D(T,L)\leqslant C(K)e^{\frac{K}{T}}.$$

Together with the known lower bound, this predicts the sharp small-time exponential rate for the cost of boundary controls of the heat equation. The statement is presented as conjectured in the source, with no resolution supplied there.