Details

The conjecture has been verified for all rectangles with as shown at [2]. The method there uses Young tableaux theory, which gets very slow above size 380.

At [3], many squared rectangles are available that were built with the method of electrical resistance on a graph. Kirchhoff's first law is that except at a pole, the sum of the currents for any node is zero. A given graph leads to a squared rectangle, where the horizontal edges correspond to nodes and squares correspond to edges. For each horizontal edge, the sum of the squares directly over an edge squares the sum of squares directly under the edge. Perusing the squared rectangles available yielded many possible counterexamples to the minimal squaring conjecture.

The rectangles at [4] are based on planar 3-connected graphs. For 6 to 13 edges, there are 1, 0, 1, 2, 2, 4, 12, 22 such graphs ([5, A002840]). But it is possible to build rectangles from planar 2-connected graphs that have a count of 3, 7, 15, 39, 106, 316, 1026, 1643 ([5, A289471]) for 6 to 13 edges. The rectangles based on 2-connected graphs always feature two neighboring squares of the same size.

References

[1] E. Pegg Jr. "Oblongs into Minimal Squares." Mathematics Stack Exchange. (Jul 11, 2017) math.stackexchange.com/questions/2057290/oblongs-into-minimal-squares.

[2] B. Felgenhauer. "Filling Rectangles with Integer-Sided Squares." (Jul 11, 2017) int-e.eu/~bf3/squares.

[3] S. Anderson. "Squaring.Net 2016." (Jul 11, 2017) www.squaring.net.

[4] Wolfgang. "Tiling a Rectangle with the Smallest Number of Squares." MathOverflow. (Jul 11, 2017) mathoverflow.net/questions/116382/tiling-a-rectangle-with-the-smallest-number-of-squares.

[5] The OEIS Foundation. The Online Encyclopedia of Integer Sequences. A002840, A289471. oeis.org.