I just found this citation, not cited in the AMS Notices paper (but I cannot yet access the Israel J Math paper itself):
Schramm, Oded. "Square tilings with prescribed combinatorics." Israel Journal of Mathematics 84, no. 1-2 (1993): 97-118. DOI.
Abstract. "Let $T$ be a triangulation of a quadrilateral $Q$, and let $V$ be the set of vertices of $T$. Then there is an essentially unique tiling $Z=(Z_v: v ∈ V)$ of a rectangle $R$ by squares such that for every edge of $T$ the corresponding two squares $Z_u, Z_v$ are in contact and such that the vertices corresponding to squares at corners of $R$ are at the corners of $Q$. It is also shown that the sizes of the squares are obtained as a solution of an extremal problem which is a discrete version of the concept of extremal length from conformal function theory. In this discrete version of extremal length, the metrics assign lengths to the vertices, not the edges."
Because I cannot access the paper, it is unclear to me if this is the source. But it seems like it might be.