 Solution

Couple two configurations differing at one vertex $u$ by choosing the same update vertex and the same uniform random number for the heat-bath update. Updating $u$ removes the disagreement. Updating a nonneighbor of $u$ cannot create one. At a neighbor $v$, the two conditional plus-spin probabilities differ by at most $\tanh \beta$, using the supplied identity. Since $u$ has at most $\Delta$ neighbors, the expected Hamming distance after one step is at mostThe Hamming diameter is $n$, so the Path coupling theorem gives