An unusually-coupled simple random walk

For motivation see section 3.3 of A lecture on the averaging process.

Consider continuous-time simple symmetric random walk on the integers; that is, transition rates \(q(i,i+1) = q(i,i-1) = 1/2\). Now let \( (X_1(t), X_2(t)), 0 \le t < \infty) \) be two such processes coupled as follows.
(i) From a state \( (i,j) \) with \( |j - i| \ge 2 \) they move independently.
(ii) From a state \( (i,i+1) \) they make each of the following moves at rate \( 1/2 \): \[ (i,i+1) \to (i-1,i+1); \quad (i,i+1) \to (i,i+2) \] and they make each of the following moves at rate \( 1/4 \): \[ (i,i+1) \to (i+1,i+1); \quad (i,i+1) \to (i,i); \quad (i,i+1) \to (i+1,i). \]
(ii) From a state \( (i,i) \) they make each of the following moves at rate \( 1/4 \): \[ (i,i) \to (i,i+1); \quad (i,i) \to (i+1,i+1); \quad (i,i) \to (i+1,i) \] \[ (i,i) \to (i,i-1); \quad (i,i) \to (i-1,i-1); \quad(i,i) \to (i-1,i) \]

Intuitively, the two component processes are "independent to first order". From duality results in the paper above we know \( \mathbb{P}(X_1(t) = i, X_2(t) = i) \ge (\mathbb{P}(X_1(t) = i))^2 \) and the problem is to find the order of magnitude of the "second order" quantity \[ v(t) := \sum_i ( \mathbb{P}(X_1(t) = i, X_2(t) = i) - (\mathbb{P}(X_1(t) = i))^2 ) \] as \(t \to \infty\) .