Let $p$ be a prime number, $K$ a $CM$ field containing a primitive $p^(n)th$ root of unity and k the maximal real subfield of $K$. We show that a special case of a conjecture of Solomon gives rise to a (conjectural) congruence mod $p^n$, relating certain $S$-units of $k$ to the principal semi-local units of $K$, via the use of local Hilbert symbols. We will discuss the progress made in proving this conjecture and (briefly) some computational verifications.