Generalization of Lévy's continuity theorem for nuclear spaces

There is a partial result due to Boulicaut (1973), which states

Theorem: Let $E$ be a separable metrizable Hausdorff locally convex topological vector space. Then $E$ is nuclear if and only if for every sequence $\{\mu_n\}$ of tight probability measures, weak convergence to a tight probability measure $\mu$ is equivalent to the pointwise convergence of the characteristic functions of $\mu_n$ to the ch. f. of $\mu$.

This makes characteristic function(al)s more useful than in separable Banach spaces, where they are only used for uniqueness but not weak convergence.

In fact both theorems are true for probability measures on $\mathcal{D}'$ and $\mathcal{S}'$ and they were proved before in the thesis of Xavier Fernique. The paper that came out of it is: "Processus linéaires, processus généralisés". Annales de l'institut Fourier, 17 no. 1 (1967), p. 1-92.

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Update: A new reference on the Lévy-Fernique continuity theorem for $\mathcal{S}'$ is https://arxiv.org/abs/1706.09326 (re MathNovice's comment below: this one is English!)