Free product of Demushkin groups as absolute Galois group

We prove that a free profinite (pro-$p$) product over a set converging to 1 of countably many Demushkin groups of rank ${\mathrm{\aleph }}_0$, $G_i$, that can be realized as absolute Galois groups, is isomorphic to an absolute Galois group if and only if ${\log }_{p}q(G_i)\to \mathrm{\infty }$.