Double subscript followed by superscript gives unpleasant result

While I am not generally a fan of making a symbol-style smaller than the scriptscriptstyle, in this case, one can almost think of nu^dag as its own glyph. So, what I do here is to define a nudagref in textstyle that has is essentially nu^dag, with the dag smashed, so that it has the dimensions of nu. Then, when I want to use it in smaller math styles, I scale the nudagref down to the size of nu in the smaller math style.

In the MWE, the first line shows dag and nudag in all three styles. Then I show what the OP had shown. Finally, I employ nudag in the final line. One downside is that the nu of nudag is a scaled textstyle version, and thus has a slightly different shape than the smaller style nu. See ALTERNATIVE below for a correction to this issue.

documentclass[]{report}
usepackage{mathrsfs,scalerel}
newcommandnudagref{textstylenu^{mkern-1musmash{dag}}}
newcommandnudag{scalerel*{nudagref}{nu}}

begin{document}
$dagnudag_{nudag_{nudag}}$

$C_{mathscr T_{nu^{smash{dagger}}}}$ vs $C_{mathscr T_nu}$

$C_{mathscr T_{nudag}}$ vs $C_{mathscr T_nu}$
end{document}

---

ALTERNATIVE:

This version preserves the proper shape of the nu in the smaller math styles.

documentclass[]{report}
usepackage{mathrsfs,scalerel}
newcommanddagref{textstylevphantom{nu}^{mkern-1musmash{dag}}}
newcommandnudag{nuscalerel*{dagref}{nu}}
begin{document}
$dagnudag_{nudag_{nudag}}$

$C_{mathscr T_{nu^{smash{dagger}}}}$ vs $C_{mathscr T_nu}$

$C_{mathscr T_{nudag}}$ vs $C_{mathscr T_{nu}}$
end{document}