Marde\v{s}i\'c conjecture and free products of Boolean algebras

We show that for every $d\ge 1$, if $L_1,\dots ,L_d$ are linearly ordered compact spaces and there is a continuous surjection $$L_1\times L_2\times \cdots \times L_d\to K_1\times K_2\times \dots \times K_d\times K_{d+1},$$ where all the spaces $K_i$ are infinite, then $K_i,K_j$ are metrizable for some $1\le i<j\le d+1$. This answers a problem posed by S. Marde\v{s}i'c. We present some related results on Boolean algebras not containing free products with too many uncountable factors. In particular, we answer a problem on initial chain algebras that was posed by L. Baur and L. Heindorf.