May I modify the value of an exception inside a std::exception_ptr?

If I have a exception stored inside std::exception_ptr. I rethrow the exception using std::rethrow_exception, access it using catch(MyException&) and then I modify the value.

If I throw the same exception again, should I observe the modification I made?

The following code demonstrate my idea:

#include <exception>
#include <iostream>

struct MyException {
    int value;
};

int main() {
    std::exception_ptr a = std::make_exception_ptr(MyException());

    try {
        std::rethrow_exception(a);
    } catch(MyException& b) {
        std::cout << b.value << std::endl;
        b.value = 3;
    }

    try {
        std::rethrow_exception(a);
    } catch(MyException& b) {
        std::cout << b.value << std::endl;
    }
}