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In a comprehensive investigation of carbon nanotube (CNT) filled liquid and solid polybutadienes of molecular weights 2600 and 100000, respectively, we report results of thermal conductivity (κ), glass transition temperature (T_g), interfacial interaction and microstructure before and after simultaneous high-pressure and high-temperature (HP&HT) treatment. The HP&HT treatment changed polybutadiene from a liquid or solid to a highly cross-linked, ebonite-like, state. Concurrently, the microstructure changed from randomly dispersed CNTs to a web-like structure of coated and/or wrapped CNTs, with a permanent shift in their D*-band by as much as 16 cm⁻¹. Moreover, κ of the recovered state of a 2.9 wt% –COOH functionalized multi-wall carbon nanotube (MWCNT) composite increased by 34% predominantly due to an irreversible densification and a consequentially increased phonon velocity. Results prior to treatment show that single-wall carbon nanotube (SWCNT) fillers promote κ better (17%/wt%) than –SH functionalized MWCNT fillers (8%/wt%), which is accounted for by their higher aspect ratio, whereas their about twice as high κ appears to be unimportant. The SWCNTs also raise T_g slightly more than MWCNTs and, in particular, under the most densified conditions and for the high molecular weight polybutadiene, which may be due to more favorable conditions for coating/wrapping.