Python pandas Dataframe Styler : relabel_index not working with to_excel output

I'm trying to use pandas Dataframe Styler possibilies (looks very cool). My need is to output the dataframe in both Excel, pdf and HTML format.

Here is the sample code :

import pandas as pd
from IPython.core.display import display, HTML
multi_index = pd.MultiIndex.from_tuples([("r0", "rA"),("r1", "rB")], names=['Courses','Fee'])
cols = pd.MultiIndex.from_tuples([("Gasoline", "Toyoto"), ("Gasoline", "Ford"), ("Electric", "Tesla"),("Electric", "Nio")])
data=[[100,300, 900,400 ], [200,500, 300,600]]
df = pd.DataFrame(data, columns=cols,index=multi_index)
print(df)
            Gasoline      Electric     
              Toyoto Ford    Tesla  Nio
Courses Fee                            
r0      rA       100  300      900  400
r1      rB       200  500      300  600

df_styled = df.style
df_styled = df_styled.relabel_index(("Gasoil", "Gasoil", "Elec.", "Elec."), axis=1, level=0)
print(display(HTML((df_styled.to_html()))))

        Gasoil  Elec.
        Toyoto  Ford    Tesla   Nio
Courses Fee              
r0  rA  100 300 900 400
r1  rB  200 500 300 600

df_styled.to_excel("test.xlsx", engine='openpyxl') 

When output is to_html or to_pdf, the result of the Styler.relabel_index is working but with the to_excel output, column names are not altered.

I'm looking for a way to separate data from output and i'm using relabel_index to localize dataframe.

Any hints ?