Is the Post Correspondence Problem with more than two rows harder than the standard two-row variant?


The standard Post Correspondence Problem concerns tiles with two rows of symbols, and whether a tile arrangement can be made so that the sequence of the top symbols of the tiles is equal to the bottom one.

Let $ \text{n-PCP}, \text{n} > 0$ a generalization of the Post Correspondence Problem where the tiles contain $ \text{n}$ rows, and the sequences of the symbols have to be equal for all of these rows.

Obviously $ \text{1-PCP}$ is decidable (in fact it’s trivial because the answer to the problem is always true). $ \text{2-PCP}$ is the standard PCP.

But what if $ \text{n} > 2$ ? Is it harder or can it be reduced to the standard PCP (like >3-SAT being reduced to 3-SAT)?