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This paper reviews the opportunities for microgravity and Earth-based measurements of the thermal conductivity λ( t, L) of4He confined in cylindrical geometries of radius L with axial heat flow at temperatures near the bulk superfluid-transition line T λ ( P) ( t is the reduced temperature T/ T λ −1). It provides an evaluation of existing data for L=1 µm near T λ at saturated vapor pressure (SVP), and uses these to derive a scaling function for the resistivity R( t, L)=1/ λ( t, L). The purpose of future measurements over a wide range of L and of the pressure P will be to test the applicability of this function. In the present paper the scaling function is used to assess quantitatively the effect of gravity on potential Earth-based measurements. It is found that the gravity effect for R is particularly severe below T λ . For typical three-mm-high samples at SVP, values of L significantly larger than 8 µm can only be investigated fully in micro-gravity. At higher pressures the gravity effect is larger. At 30 bar, three-mm samples with L≳4 µm require microgravity for measurements below T λ ( P). Modern thermometry has sufficient resolution to permit quantitative measurements in micro-gravity of the anticipated finite-size effect for values of L as large as 50 µm. [ABSTRACT FROM AUTHOR]