In [Glas. Mat., III. Ser. 5(25), 163--170 (1970; Zbl 0195.52301)], \textit{S. Mardešić} studied continuous images of products of linearly ordered compacta and stated the following conjecture: if a product of $d$ linearly ordered compact spaces can be continuously mapped onto a product of $d + s$ infinite compact spaces $K_1, K_2, \dots , K_{d+s}$, $s\ge 1$, then there exist at least $s + 1$ metrizable factors $K_i$. The authors give a positive answer to this conjecture proving it for $s = 1$, which implies that it is true in full generality. For this purpose they use a (new) kind of dimension called free dimension. Using the method of free dimension they also answer a question by \textit{L. Baur} and \textit{L. Heindorf} [Order 14, No. 1, 21--38 (1997; Zbl 0894.06004)] about the embedding of the free product of two Boolean algebras into a special Boolean algebra called initial chain algebra.