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We reduce this problem to the problem of existence of the measures $m(n,k,d)$ defined on $\\bar{\\mathcal{P}_d}$ and with Laplace transform $(\\det s)^{-n/2}\\exp \\tr(s^{-1}w)$ where $n$ is an integer and where $w=\\mathrm{diag}(0,...,0,1,...,1)$ has order $d$ and rank $k.$ We compute $m(d-1,d,d)$ and we show that neither $m(d-2,d,d)$ nor $m(d-2,d-1,d)$ exist. 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