In an old PhD-thesis "Finite products of locally compact ordered spaces" by J. van Dalen (Vrije Universiteit, Amsterdam) from 1972, I found (I could not find a paper with the result, so far, as I have no access to a university library, but I have the thesis on my shelves) the following Corollary 12.1 (page 45):

Let $m$ be a natural number and let $X_1, \ldots, X_m$ be non-degenerate connected ordered spaces. Then $\operatorname{ind}(\prod_{i=1}^m X_i)= m$.

And corollary 12.2 , which has the same data plus the extra assumption that $\prod_{i=1}^m X_i$ is normal (to make $\operatorname{Ind}$ sensibly defined, I suppose) and concludes that $\dim(\prod_{i=1}^m X_i) = \operatorname{Ind}(\prod_{i=1}^m X_i) = m$ as well.

In the references I also found

**I.K. Lifanov**, "Dimensionality of the product of ordered continua", *Dokl. Akad. Nauk SSSR* 177 (1967), 778-781 (Sov. Math. Dokl. 8 (1967), 1500-1503)

and

**I.K. Lifanov**, "The dimension of a product of unidimensional bicompacta", *Dokl. Akad. Nauk SSSR* 180 (1968), 534-537 (Sov. Math. Dokl. 9 (1968), 648-651)

which look relevant too.