Is there any explicit application of Langlands conjecture for $\mathrm{GL}(n)$
for $n\ge 3$, to get some reciprocity laws for higher dimensional varieties or higher genus curves?

I've never found such things in articles such like "What is a Reciprocity Law?", Ana Caraiani's "Higher-dimensional reciprocity laws" or "Reciprocity laws and Galois representations: recent breakthroughs".

MSE link: https://math.stackexchange.com/questions/4890980/applications-of-langlands-for-gln-explicit-reciprocity-laws-other-than-elliptic