The Plancherel-type decomposition conjecture for positive representations

Let $G_q$ be the quantized group associated with the split real simply-laced quantum group, and let $L^2(G_q)$ carry the left and right regular representations of $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$. A direct-integral decomposition means a decomposition of the Hilbert space into a direct integral of representation spaces.

Plancherel-type decomposition conjecture. The space $L^2(G_q)$ is decomposed into a direct integral of tensor products of the positive representations under the left and right regular representations of $U_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})$.

For $U_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$, an analogous decomposition is known, while the statement for general simply-laced $\mathfrak{g}$ is posed as a natural question and remains open.