We give a proof of the consistency of Dasgupta's conjectural $p$-adic formula for Gross-Stark units with Dasgupta and Spiess's alternative conjectural formula for these units. We give details of the proof when $F$ is a totally real number field of degree 2, which had been previously proven by Dasgupta and Spiess. We present work towards proving the case for a general totally real number field. Finally, we give a proof when $F$ is a totally real number field of degree 3.