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Adjacent characters (and objects and terrain markers) occupy adjacent squares, as shown in Figure 5. Squares are not adjacent if they are on the opposite sides of terrain that blocks movement (see Terrain, p.11). A character is never adjacent to the square it occupies, and a character is never adjacent to itself.
The rim is in the elevated terrain. So a square of grounded terrain can't be adjacent to the rim, since the squares are not adjacent.
However, the area of elevated terrain itself (as game effect, not as individual squares) is most certainly adjacent to the grounded terrain. Otherwise, we wouldn't know where the blocked movement would start.
Again, this would be more clear if we had a game term that meant "next to but not actually adjacent". We don't, so we have to connect some dots like this instead.