This does not answer the question, but let me argue that one can add such an injective homomorphism (for the ground-model degrees) by forcing. 

The argument relies on the fact that the Turing degree order is universal for countable orders, and indeed for countable upper semi-lattices. For example, see 

 - Antonio Montalbán, [Embeddings into the Turing degrees](https://math.berkeley.edu/~antonio/papers/Tdegrees.pdf), 2007. 

Every countable upper semi-lattice embeds into the Turing degrees.

Fix a copy of a countably universal partial order inside $\mathcal{D}$, one for which every countable order embeds into it by a back-and-forth argument.

Consider now the forcing extension $V[G]$ in which the continuum and hence also the set of Turing degrees $\mathcal{D}$ is made countable. By performing the back-and-forth argument in $V[G]$, but using the now-countable structure of ground model degrees $\mathcal{D}^V$, we can map $\mathcal{D}^V$ into itself in an order-preserving manner, realizing the embedding in $V[G]$.