Does the $K^1$-group of a complete flag variety vanish?

MathOverflow Asked by Quin Appleby on January 1, 2022

For $U(n)$ the Lie group of $n times n$ unitary matrices, and $T^n$ its maximal torus subgroup, the homogeneous space

is called the complete flag variety of order $n$. For the special case of $n=2$ this gives the sphere $S^2$. In this special case the topological $K$-theory group $K^1$ vanishes, i.e.
K^1(S^2) = 0.

Is this true for higher complete flag varieties? In explicit form, is it true that
K^1(U(n)/T^n) = 0, ~~ forall n?

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