Filed feature request to each of starship and oh-my-posh:

starship/starship#4835
JanDeDobbeleer/oh-my-posh#475

The common problem is that neither of them support TMUX coloring and styling. For both to integrate with TMUX statusline, they would have to build a converter from ANSI to TMUX format.

Thus, instead of re-inventing wheels, maybe it's a better idea for TMUX to natively support ANSI coloring/styling (or at least a converter command). This would immediately enable many projects like starship and oh-my-posh to be able to integrate with TMUX.