Tensor-product conjecture for regular positive representations

Let $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ be a quantized enveloping algebra of a split real form, and let regular positive representations mean the irreducible regular positive representations considered in the source. Their tensor product is formed using the coproduct, equivalently by amalgamating the corresponding representation data.

Tensor-product conjecture. The tensor product of regular positive representations is again regular, and it can be decomposed into a direct integral of regular positive representations.

The expected statement is motivated by the quiver realization: tensor products correspond to amalgamating basic quivers, with Chevalley-generator actions given by concatenated telescopic paths. The source states this as an expectation, and no resolution is supplied.