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Wondering about the question I posted on Friday (on StackExchange, no satisfactory answer so far!), I looked further at the special case of the gamma distribution I suggested at the end. Starting from the moment conditions,

$\dfrac{\alpha_{11}}{\beta_1} = \mu_1\,,\quad \dfrac{\alpha_{11}}{\beta_1^2} = \sigma_1^2\,,$

$\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2} = \mu_2\,,\quad \dfrac{\alpha_{21}^2\sigma^2_1}{\beta_2^2}+\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2^2} = \sigma^2_2\,,$

and

$\dfrac{\alpha_{21}(\sigma^2_1+\mu_1^2)+\alpha_{22}}{\beta_2} = \sigma_{12}+\mu_1\mu_2$

the solution is (hopefully) given by the system

$\begin{cases} \beta_1 =\mu_1/\sigma_1^2&\\ \alpha_{11}-\mu_1\beta_1 =0&\\ \alpha_{22} = \mu_2\beta_2 - \alpha_{21}\mu_1&\\ \alpha_{21} = \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)}{\sigma^2_1+\mu_1^2- \mu_1}\beta_2\\ \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)^2}{(\sigma^2_1+\mu_1^2- \mu_1)^2} \sigma_1^2 + \dfrac{\mu_2}{\beta_2} = \sigma^2_2&\\ \end{cases}$

The resolution of this system obviously imposes conditions on those moments, like