= 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 most
$$
1-\frac1n+\frac{\Delta\tanh\beta}{n}
=1-\frac{1-\Delta\tanh\beta}{n}.
$$
The Hamming diameter is $n$, so the <Path coupling theorem> gives
$$
d(t)\leq n\left(1-\frac{1-\Delta\tanh\beta}{n}\right)^t.
$$
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