Status on Naoki Katoh's "Rectangle Wiring Problem" (minimum length tree to cover a partitioned rectangle)?
This problem has been studied since 2002, under the name minimum-length corridor problem. 

The problem is known to be NP-complete[1,2], but there is a constant factor polynomial time approximation algorithm[3].

There is more work on this problem, which you can find by searching the citation network of the papers below.



[1]: GONZALEZ-GUTIERREZ, Arturo; GONZALEZ, Teofilo F. Complexity of the minimum-length corridor problem. Computational Geometry, 2007, 37.2: 72-103.

[2]: BODLAENDER, Hans L., et al. On the minimum corridor connection problem and other generalized geometric problems. Computational Geometry, 2009, 42.9: 939-951.

[3]: GONZALEZ-GUTIERREZ, Arturo; GONZALEZ, Teofilo F. Approximation Algorithms for the Minimum-Length Corridor and Related Problems. In: CCCG. 2007. p. 253-256.