Let $F$ be a totally real number field, and let $\mathfrak{p}$ be a finite prime of $F$ such that $\mathfrak{p}$ splits completely in the finite abelian extension $H$ of $F$ . Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a $\mathfrak{p}$ -unit $u$ in $H$ with absolute values at the places above $\mathfrak{p}$ specified in terms of the values at zero of the partial zeta functions associated to $H/F$ . This conjecture is an analogue of Stark's conjecture, which Tate called the Brumer-Stark conjecture. Gross [12, Conjecture 7.6] proposed a refinement of the Brumer-Stark conjecture that gives a conjectural formula for the image of $u$ in $F_\mathfrak{p}^\times/\widehat{E}$ , where $F_\mathfrak{p}$ denotes the completion of $F$ at $\mathfrak{p}$ and $\widehat{E}$ denotes the topological closure of the group of totally positive units $E$ of $F$ . We present a further refinement of Gross's conjecture by proposing a conjectural formula for the exact value of $u$ in $F_\mathfrak{p}^\times$