Case Insensitive String Matching in Awk

Assume a multi-line text file file in which some lines start with whitespaces.

$ cat file
foo Baz
  baz QUX
    QUx Quux
BaZ Qux
BazaaR

Further assume that I wish to convert all those lines that start with a keyword (e.g. "baz") to lowercase letters, irrespective if (a) that keyword is written in lower- or uppercase letters (or any combination thereof) itself, and (b) that keyword is preceeded by whitespaces.

$ cat file | sought_command
foo Baz        # not to lowercase (line does not start with keyword)
  baz qux      # to lowercase
    QUx Quux
baz qux        # to lowercase
BazaaR         # not to lowercase (line does not start with keyword, but merely with a word containing the keyword)

I believe that awk is the tool to do it, but I am uncertain how to implement the case-insensitivity for the keyword matching.

$ cat file | awk '{ if($1 ~ /^ *baz/) print tolower($0); else print $0}'
foo Baz
  baz qux
    QUx Quux
BaZ Qux       # ERROR HERE: was not replaced, b/c keyword not recognized.
BazaaR

EDIT 1: Adding IGNORECASE=1 appears to resolve the case-insensitivity, but now incorrectly converts the last line to lowercase.

$ cat file | awk '{IGNORECASE=1; if($1~/^ *baz/) print tolower($0); else print $0}'
foo Baz
  baz qux
    QUx Quux
baz qux
bazaar       # ERROR HERE: should not be converted to lowercase, as keyword not present (emphasis on word!).