Regex for non-consecutive uppercase with lowercase

See more:

regular-expression

C++14

std::string s("AAA");
std::smatch m;
std::regex e("(?=.{3,}[A-Z])(?=.{0,}[a-z]).*");
output = std::regex_search(s, m, e);

Here it should have 3 or more uppercase letters and zero or more lowercase letter.
However the output is zero meaning it fails. When I try replace zero with 1 and
s("AAA4") it works fine.

So now I want it to allow zero or more but it seems like it is not accepting zero
I even tried quantifier (*) which is equivalent to {0,} still not working.

What I have tried:

.

I also tried [A-Z]{3,}[a-z]* 
but this will allow:
s("AAA") and s("AAAb") but now s("AbAA")
as it will allow only if upper are consecutive.