Numerical Stairs Description

Numerical Stairs is a puzzle original to this blog to the best of my knowledge. The grid is partitioned into a number of rooms. The object is to write exactly one number in each room. Some numbers may be given; note exactly where the number is written is inconsequential, and the number is always considered to span the whole room. All numbers must be in the range of 1 to 9. No two adjacent rooms may contain the same number. Furthermore, any two adjacent rooms whose numbers are consecutive have a path between them. Combined, these paths must form a single closed loop that passes through every room of the puzzle and never branches or intersects itself. In particular, every room must be adjacent to exactly two rooms with a consecutive number, and two rooms without a path between them must have numbers that differ by at least two. The solution is unique.

Below is an example puzzle and its only solution. The loop formed by the paths is drawn in light red for clarity, although you only need to write the numbers in to solve a puzzle.