We have recently observed nonlinear heat transport within 30 nK of the superfluid transition temperature using heat flux, Q, in the range 0.1 < Q < 2 erg/(s cm2). While Haussmann and Dohm (HD) accurately predict the initial departure of the thermal conductivity, κ, from the linear response region, κ is greater than expected very close to Tλ. We anticipate that the nature of the thermal conductivity's nonlinearity may depend upon Earth's gravity in the low heat flux limit (Q < 0.5 erg/(s cm2)). Comparison of our data to similar data to be taken in a microgravity laboratory will provide an experimental determination of the effect of gravity on nonlinear heat transport near the superfluid transition. The microgravity measurements will also permit the first experimental test of theories that do not consider gravitational effects, such as those by HD.