  We study the algebra of upper triangular matrices endowed with a group
grading and a homogeneous involution over an infinite field. We compute the
asymptotic behaviour of its (graded) star-codimension sequence. It turns out
that the asymptotic growth of the sequence is independent of the grading and
the involution under consideration, depending solely on the size of the matrix
algebra. This independence of the group grading also applies to the graded
codimension sequence of the associative algebra of upper triangular matrices.
