The decomposition uniqueness for infinite Cartesian products

It is well known that the finite product of locally connected curves has the decomposition uniqueness property. It is natural to ask whether the same holds for infinite products. In general, this isn't the case - the Hilbert cube is homeomorphic to the countable infinite product of triods. We prove that if X is a product of locally connected curves then X has the decomposition uniqueness property if only finitely many of the factors are locally dendrites. The last condition is not necessary. It has been shown by Eberhart that the infinite torus has the decomposition uniqueness property.