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, 2007.

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

Fix a copy of the countably universal partial order inside $\mathcal{D}$. 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 $V[G]$.

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, 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 $V[G]$.

This does not answer the question, but kindly let me mention merely that the Turing degree order is known to be universal for countable orders, and indeed for countable upper semi-lattices. For example, see

• Antonio Montalbán, Embeddings into the Turing degrees, 2007.

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

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, 2007.

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

Fix a copy of the countably universal partial order inside $\mathcal{D}$. 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 $V[G]$.