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

call return the key 7, which is expected. Then the between and iterator should return 6, as it should search between key 4 and key 7 and the max value is 6 for key 7.
But it's returning the last for some reason which I can't 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;

}