Product decomposition conjecture for densities on the torus

Let $\mathrm{Dens}({\mathbb T}^n)$ denote the smooth normalized densities on the $n$-torus, and let $\mathrm{Diff}_{\mathrm{d}x}({\mathbb T}^n)$ denote the volume-preserving diffeomorphisms. For $\mu\in\mathrm{Dens}({\mathbb T}^n)$, seek a volume-preserving change of coordinates.

Product decomposition conjecture. Given $\mu\in\mathrm{Dens}({\mathbb T}^n)$, there exist $\eta\in\mathrm{Diff}_{\mathrm{d}x}({\mathbb T}^n)$ and functions $\lambda_k\colon S^1\to{\mathbb R}^{+}$ such that

$$(\eta^*\mu)(x_1,\ldots,x_n)=\lambda_1(x_1)\lambda_2(x_2)\cdots\lambda_n(x_n)\,\mathrm{d}x.$$

This would provide an infinite-dimensional analogue of spectral decomposition for symmetric matrices by representing a density, up to a volume-preserving coordinate change, as a product of one-variable factors. The source does not state a resolution.