Yep - tone mapping Rec 2020 HDR to Rec 709 SDR is a requirement for playback of Rec 2020 HDR content nicely on Rec 709 SDR TVs.
(Rec 709 vs Rec 2020 is the colour gamut, SDR vs HDR is the dynamic range - both need to be handled)
However it's important not to assume it's a UHD 2160p vs HD 1080p thing - there are UHD TVs that need tone mapping as they are Rec 709 SDR only (early 4K TVs weren't Wide Colour Gamut or HDR), and there are 1080p HD TVs that don't need tone mapping, just downscaling (as they are Rec 2020 HDR compatible)
Ok thanks for your help.