Product-state conjecture for tensor-product unitary operators

Let $U$ be the unitary operator linking two observables, with

$$U=U_1\otimes U_2.$$

For a state $\rho$, its entropy pair means the two entropies associated with the two observables.

Product-state conjecture. For every state $\rho$, there exists a product state $\rho_1\otimes\rho_2$ with the same pair of entropies.

If true, this would reduce the minimal-entropy curve for product-form operators in dimension $d=d_1d_2$ to tensor products of states parametrizing the corresponding curves in dimensions $d_1$ and $d_2$.