Squaring a square and discrete Ricci flow Is this a theorem?

Every $3$-connected planar graph $G$ may be represented as
a tiling of a square by squares,
one square per node of $G$, with nodes connected in $G$
corresponding to tangent squares.




I ask because the cited AMS Notices paper1
suggests that this might
be a theorem, offering it as an
example of "discrete Ricci flow" and a
"generalization of circle packing."
But no explicit reference is provided.
There are so many unresolved questions concerning square dissections
that it would be interesting to see if this square packing
result (if indeed it holds in some form) could address some
of the unknowns.

1Gu, Xianfeng, Feng Luo, and Shing Tung Yau.
"Computational conformal geometry behind modern technologies."
Notices of the American Mathematical Society 67, no. 10 (2020): 1509-1525.
PDF download.
 A: I don't know about having one vertex per square, but there is a similar very interesting construction with edges at squares. It does not answer your question but it will still surely interest you.
Specifically, given a rectangle of height $a$ and width $b$ tiled by squares, you can put a vertex on each horizontal line segment and one edge per square connecting its top and bottom sides.
Now, label all edges by the side lenght of the corresponding square and orient them from top to bottom.
By seeing the labels as both currents and voltages, note that the resulting graph satisfy Kirchoff two laws, except at the top and bottom vertices. This is fixed by adding an edge from the bottom vertex to the top one with current $b$ and voltage $-a$: we then obtain a $3$-connected planar graph satisfying Kirchoff laws everywhere.
Conversely, given a $3$-connected planar graph with a specified edge $e$, if we impose some voltage $V_1$ on $e$ and resistance $1$ on all other edges, we can solve for all the currents and voltages with Kirchoff laws and this will give us the graph of a rectangle dissection into squares. If the equivalent resistance happen to be $1$, this will mean $a=b$, i.e. the big rectangle is a square.
For more details, this idea is described in chapter 11 of Richard Stanley's Algebraic combinatorics.
A: What does "tangent" mean? If two squares touch only at a vertex, are they tangent? According to the diagram provided, tangent seems to mean that two squares should meet along some (nontrivial) segment of their edges (for instance 23 and 19 meet at a vertex, even though they do not have an edge in common).
If meeting only at a vertex is not counted as being tangent, then  the edge graph of a tetrahedron is a counterexample to the proposed theorem (it is not possible to tile a square into 4 squares with each square touching all other squares along a segment of an edge).
A: 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.
A: My question is answered in Lovász's book:

Lovász, László. Graphs and Geometry. Vol. 65. American Mathematical Soc., 2019.

p.82:

Theorem 6.2. Every planar map in which the unbounded country is a quadrilateral,
all other countries are triangles, and is not separated by a $3$-cycle or a
$4$-cycle, can be represented as a resolved tangency graph of a square tiling of
a rectangle.

Concerning corner touching, Lovász says that,
in the case where four squares share a vertex,

we can specify arbitrarily one diametrically opposite pair as
"infinitesimally overlapping," and connect the centers of these two
square[s] but not the other two squares.
We call this a resolved tangency graph of the family of squares.

Indeed the source is Schramm 1993.
