why std::max_element over std::map within some lower_bound not working?

I am not sure why the following snippet is not working: The lower_bound call return the key 7, which is expected.Then the std::max_element between begin() and lower_bound() iterator should return 6, as it should search between key 4 and key 7 and the max value is 6 for key 7. But its returning the last pair(15, 12) for some reason which I cant figure out.

bool cmp(const std::pair<T, T>& p1, const std::pair<T, T>& p2) 
{
    return p1.second < p2.second;
}

int main()
{
    std::map< T, T, std::greater<T> > store;
    std::map< T, T, std::greater<T> >::iterator found_max, lower;

    store[ 4 ] = 2;
    store[ 7 ] = 6;
    store[ 10 ] = 2;
    store[ 15 ] = 12;
    lower = store.lower_bound( 8 );

    printf("%ld %ld\n", lower->first,lower->second);

    found_max = std::max_element(store.begin(), lower, cmp);
    printf("%ld %ld\n", found_max->first,found_max->second);

    return 0;
}