= Solution
Use the standard interpretation that the counts are conditionally <independent> given the parameters, with mutually <independent> stated priors. Let $a_2=\alpha_2$, $b_2=\beta_2$, and define
$$
T=\sum_{i,j,t}x_{ijt},\qquad
R=\sum_{i,t}x_{i2t},\qquad
C=\sum_{i,j}x_{ij2},\qquad
E=I(1+e^{a_2})(1+e^{b_2}).
$$
Under the reference constraints, the <Poisson distribution> <likelihood>, up to data-only factors, is
$$
L(\theta,a_2,b_2)=\theta^T\exp\{Ra_2+Cb_2-\theta E\}.
$$
Multiplying by the shape-rate gamma prior gives the <Poisson-gamma conjugacy with unequal exposures>
$$
\boxed{\theta\mid a_2,b_2,\mathbf x
\sim\Gamma(a+T,b+I(1+e^{a_2})(1+e^{b_2})).}
$$
Here $b$ is a rate, matching the <probability density function> convention on the PDF's first page. The factor $I$ represents the number of replicates per cell, not the total count.
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