That is quite a bit of thermal mass to heat up..long after the polar caps have melted away.
4000m ROVs and AUVs:
Several unmanned research submersibles are capable of diving to 4000 meters (and deeper).
Examples:
Nereus (Woods Hole Oceanographic Institution) Reached 10,902 m in the Mariana Trench before being lost in 2014. It could operate in tethered ROV mode or as an autonomous AUV.
REMUS 6000 (WHOI/Kongsberg) Autonomous Underwater Vehicle (AUV) rated to 6000 m. Used in searches like Air France Flight 447.
ROV Jason (WHOI) A remotely operated vehicle rated for 6500 m. Frequently deployed for deep-ocean science.
ROV Isis (UK NOC) Operates down to 6500 m.
HROV Hybrid Remotely Operated Vehicle WHOI class designed for 40006000 m range work.
Chinese ROVs (e.g., Qianlong AUV series, Haima ROV) Several units built for 4500 m and beyond.
ROV Victor 6000 (Ifremer, France) Depth rating 6000 m.
4000 m is well within the operating envelope of many unmanned research subs, both AUVs (autonomous) and ROVs (remotely operated).
It is time to map in earnest, the almost 75% of the World's ocean bottom instead of blowing national treasure on establishing a lunar base.