Infiniteness of A_∞-types of gauge groups

Let G be a compact connected Lie group and let P be a principal G-bundle over K. The gauge group of P is the topological group of automorphisms of P. For fixed G and K, consider all principal G-bundles P over K. It is proved in Crabb and Sutherland [Proc. London Math. Soc. (3) 81 (2000) 747-768] and Tsutaya [J. London Math. Society 85 (2012) 142-164] that the number of A_n-types of the gauge groups of P is finite if n < ∞ and K is a finite complex. We show that the number of A_∞-types of the gauge groups of P is infinite if K is a sphere and there are infinitely many P.