In this paper by Nordenstam, it is shown that a certain interlacing particle process that arises from uniformly random Aztec diamond tilings is amazingly similar to Warren's process. One of the results (theorem 1.1) is that each level converges to a Dyson Brownian. The main conjecture is that the full process converges to Warren's process.
I have two questions concerning this.
Question 1: What is the status of of the main conjecture in Nordenstam's paper?
Question 2: This is an extremely basic misunderstanding on my part: The Aztec diamond particle process of section 2 of Nordenstam's paper is defined in such a way that each line of particles $X^k(t)=(X_1^k(t),\cdots, X_k^k(t))$ either stays still, or jumps forward one step at any given time. Given that the motion is only in one direction, how is it possible that this becomes Dyson brownian motion, which can go both up and down?