  Let $F$ be a totally real number field. Dasgupta conjectured an explicit
$p$-adic analytic formula for the Gross-Stark units of $F$. In a later paper,
Dasgupta-Spiess conjectured a cohomological formula for the principal minors
and the characteristic polynomial of the Gross regulator matrix associated to a
totally odd character of $F$. Dasgupta-Spiess conjectured that these
conjectural formulas coincide for the diagonal entries of Gross regulator
matrix. In this paper, we prove this conjecture when $F$ is a cubic field.
