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