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The aim of this short paper is to explore a new connection between aconjecture concerning sharp boundary observability estimates for the 1-D heatequation in small time and a conjecture concerning the cost ofnull-controllability for a 1-D convection-diffusion equation with constantcoefficients controlled on the boundary in the vanishing viscosity limit, inthe spirit of what is done in [Pierre Lissy, A link between the cost of fastcontrols for the 1-D heat equation and the uniform controllability of a 1-Dtransport-diffusion equation, C. R. Math. Acad. Sci. Paris, Volume 352, 2012].We notably establish that the first conjecture implies the second one as soonas the speed of the transport part is non-negative in the transport-diffusionequation.

An application of a conjecture due to Ervedoza and Zuazua concerning the  observability of the heat equation in small time to a conjecture due to Coron  and Guerrero concerning the uniform controllability of a convection-diffusion  equation