Nope, so this is what you do.
Just like in older version of AutoCAD, layer 0 is a special layer, but it works a little different in C3D.
So what you do is change the layer of the label style to 0 and everything to byblock. this allows the entire note to take on the properties of the layer you change it to after the note is added.
R10 - 2020 ACAD
2008 - 2020 Civil 3D
2014 - 2020 Plant 3D
2014 - 2020 Revit
V8i - V8i SS4 Microstation
Infraworks
2018-2020 Inventor
2020 Navisworks