If $K_1$ and $K_2$ are subfields of a pre-chosen $\overline{\mathbb{Q}_p}$, and if they're both unramified at $p$, and $[K_1:\mathbb{Q}_p]=[K_2:\mathbb{Q}_p]$, does that imply that $K_1=K_2$?
My intuition says that this is true because all that's happening is that we're extending the residue field, and there's only one way to do that if we fix the degree of the extension. But that's not a proof...