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In this work, we study the thermal conductivity of densified soda lime borosilicate glasses with varying B2O3/SiO2 ratio. Densification is induced by hot compression up to 2 GPa at the glass transition temperature. We find that the structural and mechanical properties of the glasses exhibit a similar response to hot compression as other oxide glasses, including increasing density, elastic moduli, and fraction of four-coordinated boron across the full compositional range. Generally, we find that thermal conductivity increases upon densification, but with a pronounced composition dependence, as silica-rich glasses exhibit only a minor increase (~8-10%) while borate-rich glasses exhibit a significant increase (>50%). We rationalize these variations in terms of topological constraint theory by showing a connection between the contribution of propagative vibrational modes to heat transfer and the volumetric constraint density across both as-made and densified samples. These findings thus provide insights into the linkages between structure and thermal conductivity.