I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question will be explained after a (unfortunately) long list of definitions (which can be considered as axioms).
A $n$-mode Gaussian state $\rho_g$ is a trace class operator in $L^2(\mathbb{R}^n)$ where canonical position and momentum operators $q_1,q_2,\cdots,q_n;p_1,p_2,\cdots,p_n$ (satisfying CCR) has a normal distributions on a real line. For such states it is convenient to work on $2n \times 2n$ covariance matrix $S$. By Heisenberg uncertainty, it can be shown that \begin{equation} S+\frac{\imath}{2}J_{2n}\geq 0;\quad \text{ where }~~ J_{2n}= \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix}. \quad (1) \end{equation} $I_n$ is identity matrix.
A Gaussian channel is a (trace preserving - for trace class operators) completely positive map $L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$, such that for every input Gaussian state the output is Gaussian. In terms of covariance matrix this gives a pair $(A,B)$ where $B\geq0$ and a map $S \mapsto A^tSA +\frac{1}{2}B$, such for every $S$ satisfying $(1)$ so do the right hand side.
A result of Heinosaari, Holevo and Wolf gives that if the pair $(A,B)$ satisfies \begin{equation} B+\imath(A^tJ_{2n}A -J_{2n}) \geq 0, \quad(2) \end{equation} then it defines a Gaussian channel. This is obvious and directly follows from equation $(1)$.
Any unitary operator $U$, which preserves Gaussianity (i.e. for all $n$-mode Gaussian states $\rho_g$ the output $U^* \rho_g U$ is Gaussian) gives rise to a $2n \times 2n$ symplectic matrix $L$ which acts on covariance matrices in the way $S\mapsto {L^{-1}}^t S L^{-1}$. We call such unitaries as Gaussian unitary (Second quantization).
If a Gaussian channel give as in item 2 satisfies equation $(2)$ of item 3, then there is a unitary dilation of the channel as given here. This means given $(A,B)$ satisfying $(2)$, there is a $L\in Sp(2(n+l))$, where dimension $l$ has to be chosen suitably. Conversely, if there is a Gaussian unitary dilation of any channel, then it must satisfies equation $(2)$. This again is easy to see.
My question: Given any Gaussian channel i.e. given $(A,B)$, and without assuming equation $(2)$, can we show that there is a Gaussian unitary dilation? Notice that, this question is weaker the one in the last reference because we are not assuming $(2)$. We are only assuming that given $(A,B)$, for each $S$ satisfying equation $(1)$ the output $A^tSA + \frac{1}{2} B$ also satisfies $(1)$. Any completely positive map has unitary dilation, but can we show that for Gaussian channels that unitary is Gaussian so that by second quantization we get the corresponding symplectic matrix $L$. I could not do it from the above mentioned materials. May be I am missing some point(s).
Sorry for making a long statement. I am not sure whether this is the correct forum to ask this question. I request moderators to shift/ copy the question in the appropriate forum, if necessary. Advanced thanks for any comment/suggestion.