original request:
https://groups.google.com/a/chromium.org/d/topic/chromium-hterm/ncWSu-hV7r8/discussion

updated info:
https://groups.google.com/a/chromium.org/d/msg/chromium-hterm/ncWSu-hV7r8/P_X5JtXdBQAJ
https://github.com/kovidgoyal/kitty/blob/master/protocol-extensions.asciidoc

The new SGR 4:3 (\e[4:3m) attribute, strictly with a colon as separator, was introduced to start a curly underline.

In the mean time, 4:0, 4:1 and 4:2 were also added as aliases for the standard 24 (turn off all kinds of underlining), 4 (single underline) and 21 (double underline), respectively.

Tthe new SGR 58 (just as you proposed in this thread) and 59 sequences specify the color of the underline, following the pattern of 38 and 39. That is, 58;5;idx for an entry of the 256-color palette, or 58;2;r;g;b for direct RGB. There's no shortcut notation for the first 16 entries (corresponding to SGR 30-37 and 90-97), use the 256-color mode with indices of 0-15 instead.

59 reverts to the default, that is, the underline's color auto-following the text color.