A tail cone version of the Halpern-L\"auchli theorem at a large cardinal

The classical Halpern-L\"auchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height $\omega$, there exist strong subtrees sharing the same level set such that tuples consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern-L\"auchli theorem at large cardinal, which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.