Consider two compact, simply connected surfaces $D$ and $D'$ in $\mathbb{R}^3$ with non-empty boundaries, and endowed with Riemannian metrics $g$ and $g'$, respectively. Does there exist a conformal mapping from $D$ to $D'$ that preserves the boundary ?
By conformal map here I mean $g'=e^{2\omega}g$ for some scalar $\omega$, and by preserving the boundary we can say that up to rotations and translations in $\mathbb{R}^3$ they (the boundaries) are the same, or equivalently (I guess) that their geodesic curvature functions are equal. It would suffice to show it when one of the surfaces is flat, I think.