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We experimentally analyze the average-power-scaling capabilities of ultrafast, thulium-doped fiber amplifiers. It has been theoretically predicted that thulium-doped fiber laser systems, with an emission wavelength around 2 um, should be able to withstand much higher heat-loads than their Yb-doped counterparts before the onset of transverse mode instability (TMI) is observed. In this work we experimentally verify this theoretical prediction by operating thulium doped fibers at very high heat-load. In separate experiments we analyze the performance of two different large-core, thulium-doped fiber amplifiers. The first experiment aims at operating a short, very-large core, thulium-doped fiber amplifier at extreme heat-load levels of more than 300 W/m. Even at this extreme heat-load level, the onset of TMI is not observed. The second experiment maximizes the extractable average-output power from a large-core, thulium-doped, fiber amplifier. We have achieved a pump-limited average output power of 1.15 kW without the onset of TMI. However, during a longer period of operation at this power level the amplifier performance steadily degraded and TMI could be observed for average powers in excess of 847 W thereafter. This is the first time, to the best of our knowledge, that TMI has been reported in a thulium-doped fiber amplifier.