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The heliosphere is bathed in Lyα radiation from both solar and stellar sources which are observed as backscattered “Lyα glow” maps and “Lyα absorption” profiles, respectively. We show that Lyα absorption profiles produced from BU “croissant-shaped” heliosphere models are consistent with HST observations and absorption profiles produced from Moscow “cometlike” heliosphere models. Under identical local interstellar medium (LISM) conditions the “croissant- like” and “comet-like” models produce nearly identical profiles for upwind target stars and similar but notably different profiles for downwind targets. These differences are due to a stronger hydrogen wall and weaker heliosheath absorption that produce a shift in the profile by 4 km/s and a 5% shallower absorption in the “croissant-shaped” model. The best agreement with HST observations is obtained using LISM conditions with higher neutral density (nH,LISM ), higher plasma density (np,LISM ), and weaker magnetic field (BLISM = 3.2 , αBV ∼ 40°) than implied by Voyager observations, implying an unresolved tension between the remote and in-situ observations. We show that for the LISM conditions with the least certainty (np,LISM , nH,LISM , TLISM , BLISM ), BLISM has the most significant effect on the structure of the hydrogen wall and Lyα profiles. Finally, we show that Lyα glow maps are also affected by the BLISM due to differences in the hydrogen wall strength and consequently the filtration of neutral hydrogen from the interstellar medium into and through the heliosphere.