= Solution
The only independent current is
$$
J(t)=\alpha P_1-\beta P_2=re^{-(\alpha+\beta)t}.
$$
Hence the total transient entropy production is
$$
\boxed{\dot S_{\rm tot}=J(t)\log\frac{\alpha P_1(t)}{\beta P_2(t)}\geq0.}
$$
Its relaxational part may be written
$$
\dot S_{\rm rel}=J\log\frac{P_1P_2^{\rm ss}}{P_2P_1^{\rm ss}},
\qquad
(P_1^{\rm ss},P_2^{\rm ss})=\frac1{\alpha+\beta}(\beta,\alpha),
$$
while the steady or housekeeping entropy production is zero. Both the current and total production vanish as $t\to\infty$.
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