How can I deduce function template arguments of type std::bitset<N>::reference?

Question

I am trying to create a function overload for std::bitset<N>::reference. I know this usually isn't useful since implicit conversions to/from bool already exist, but I believe it is necessary here (although I am open to other suggestions).

As an example of something similar to my use case, consider trying to overload the istream extraction operator for bitset references:

#include <bitset>
#include <iostream>

template<typename Ch, typename Tr, std::size_t N>
std::basic_istream<Ch, Tr> &operator>>(
    std::basic_istream<Ch, Tr> &is,
    typename std::bitset<N>::reference &ref)
{
    bool x;
    is >> x;
    ref = x;
    return is;
}

int main()
{
    std::bitset<4> X;
    std::cin >> X[2];
    std::cout << X[2];
}

This fails to compile for at least GCC and Clang since the compiler cannot deduce the size N.

Here are some ideas I have already considered which don't fully solve the problem:

• Make a more generic overload and attempt to SFINAE-out unwanted instantiations. However, I am unaware of any way to definitively test whether a given type is a bitset reference and I am worried that this could lead to ambiguous overload errors in the future.
• Use a deduction guide. However, I don't know how to obtain the value of N given only the reference type, and even if this was possible, I would prefer that the code worked in C++14.
• Modify the STL implementation. This is not possible for my use case.
• Implement my own bitset. I would prefer to use std::bitset if possible.

A comment on this question says this sort of thing is impossible in the general case. Is there a better way to do this than these options in the special case of std::bitset<N>::reference? Thanks in advance!