The integral Verma-to-Fock embedding conjecture

Let ${}_\mathbf A M(\boldsymbol a)$ be the $\mathbf A_T$-form of the Verma module generated by $|\boldsymbol a\rangle$, and let ${}_\mathbf A F(\boldsymbol a)$ be the $\mathbf A_T$-form of the Fock module of the Heisenberg algebra generated by $|\boldsymbol a\rangle$ and the operators $\widetilde P^i_n$. Let $\mathscr W_\mathbf A(\mathfrak g)$ be the integral form of the W-algebra and $\widetilde H^0_\mathbf A(\mathfrak g)$ the corresponding integral Heisenberg algebra. Integral Verma-to-Fock embedding conjecture. There is an embedding

$${}_\mathbf A M(\boldsymbol a)\longrightarrow{}_\mathbf A F(\boldsymbol a)$$

compatible with the embedding

$$\mathscr W_\mathbf A(\mathfrak g)\longrightarrow\widetilde H^0_\mathbf A(\mathfrak g).$$

This is a working hypothesis expected to follow from an appropriate definition of the integral form of the Verma module; the source does not prove it.