Unitary-similarity conjecture for Lie-group representations with identical Lie-algebra identities

Let $\rho_i:G_i\to\mathbf{U}(H)$ be faithful unirreps of connected simply connected Lie groups in a separable Hilbert space, with corresponding Lie algebras $\mathfrak{g}_i$. Let $H^{\infty}_{G_i}$ be the smooth-vector spaces and $\partial\rho_i:\mathfrak{g}_i\to L(H^{\infty}_{G_i})$ the induced Lie-algebra representations. Suppose these representations have the same identities, and suppose either (a) $G_1$ and $G_2$ are nilpotent, or (b) $G_i$ are semisimple and $\rho_i$ are discrete-series representations for $i=1,2$. Unitary-similarity conjecture. Then $\rho_1$ is unitarily similar to $\rho_2$. The surrounding text gives an analogous theorem for connected real Lie groups at the level of the associative algebras generated by the enveloping-algebra actions; the stated similarity conclusion under the two additional classes of hypotheses is not resolved in the supplied material.