Coron–Guerrero uniform controllability conjecture for convection-diffusion equations

Let $M\ne0$, $\varepsilon\in(0,1)$, and consider the transport-diffusion equation on $(0,T)\times(0,L)$,

$$y_t-\varepsilon y_{xx}+My_x=0,$$

with boundary control at $x=0$, homogeneous boundary condition at $x=L$, and initial data in $H^{-1}(0,L)$. Let $C_{TD}(T,L,M,\varepsilon)$ denote the corresponding optimal control cost. Coron–Guerrero conjecture. For given $T>0$, $L>0$, and $M\ne0$, one has

$$C_{TD}(T,L,M,\varepsilon)\longrightarrow0\quad\text{as }\varepsilon\to0^+$$

as soon as $T>L/M$ when $M>0$, and $T>2L/|M|$ when $M<0$. The conjecture proposes uniform controllability in the large-time regimes suggested by the vanishing-viscosity transport limit. The source explicitly says that the conjecture had not been decided to be true or false there; exponential lower bounds are known below the stated thresholds.