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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,

and

the solution is (hopefully) given by the system

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

So I ran a small R experiment checking when there was no acceptable solution to the system. I started with five moments that satisfied the basic Stieltjes and determinant conditions

# basically anything
mu=runif(2,0,10)
# Jensen inequality
sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1))
# my R code returning the solution if any
sol(mu,c(sig,runif(1,-sqrt(prod(sig)),sqrt(prod(sig)))))

However, not being sure about the constraints on the five moments I am now left with another question: what are the necessary and sufficient conditions on the five moments of a pair of positive vectors?! Or, more generally, what are the necessary and sufficient conditions on the k-dimensional μ and Σ for them to be first and second moments of a positive k-dimensional vector?