I am trying to understand a relation between the two theorems stated in the title. What I observed so far is that since $H^{2}(\mathbb{R},U(1))=\{e\}$, using Bargmann's theorem, we have that projective unitary representations of the additive group $(\mathbb{R},+)$ are in one-to-one correspondence with unitary representations of the additive group $(\mathbb{R},+)$, which will lead to the Schrödinger equation via Stone's theorem.

On the other hand, we have that $H^2(\mathbb{R}^{2n},U(1))=\mathbb{R}$, so the unitary projective representations of $\mathbb{R}^{2n}$ are not in one to one correspondence with its unitary representations, rather in one-to-one correspondence with the unitary representations of the Heisenberg group, a central extension of $\mathbb{R}^{2n}$. This is what we get from Bargmann's theorem.

Finally, the Stone-Von-Neumann theorem says that all the irreducible unitary representations of the Heisenberg group are unitarily equivalent to the Schrödinger representation. I stated these theorems without all their technical details, but my question is:

Can the two theorems be composed to obtain the following theorem, which, let's say, we call (Stone-Von-Neumann-Bargmann):

There exists a unique (up to unitary equivalence) strongly continuous projective representation of $\mathbb{R}^{2n}$ for all $n \in \mathbb{N}$ (including zero) on a separable Hilbert space.