OK. It seems there are results for more than 2 goods, but the results are quite weak:
Thus, if both relative prices are below the relative prices in autarky, we can rule out the possibility that both goods 1 and 2 will be imported—but we cannot rule out the possibility that one of them will be imported. In other words, once we leave the two-good case, we cannot establish detailed predictive relations saying that if the relative price of a traded good exceeds the relative price of that good in autarky, then that good will be exported by the country in question. It follows that any search for a strong theorem along the lines of our first proposition earlier is bound to fail. The most one can hope for is a correlation between the pattern of trade and differences in autarky prices.
Dixit, Avinash; Norman, Victor (1980). Theory of International Trade: A Dual, General Equilibrium Approach. Cambridge: Cambridge University Press. p. 8
OK. It seems there are results for more than 2 goods, but the results are quite weak: