Consider the Lie group of unitary matrices endowed with its bi-invariant metric $(U(n), g)$, where $g(X, Y) := \text{Tr}(Y^\dagger X)$ for any two tangent vectors $X, Y$.
Let us also consider the complex Stiefel manifold $V_k(\mathbb{C}^n) \simeq U(n) / U(n-k)$, and the complex Grassmann manifold $\mathit{Gr}_k(\mathbb{C}^n) \simeq U(n) / (U(n-k) \times U(k)) \simeq V_k(\mathbb{C}^n) / U(k)$.
It is known that we can endow the Stiefel manifold with a metric $h$ such that the projection $\pi_1 : (U(n), g) \to (V_k(\mathbb{C}^n), h)$ is a Riemannian submersion with totally geodesic fibres.
It is also known that we can endow $\mathit{Gr}_k(\mathbb{C}^n)$ with a metric $\tilde h$ such that the projection $\pi_2 : (V_k(\mathbb{C}^n), h) \to (\mathit{Gr}_k(\mathbb{C}^n), \tilde h)$ is a Riemannian submersion.
My question is: does $\pi_2$ have totally geodesic fibres?
It is clear to me that the fibres of $\pi_1$ are totally geodesic, as $U(n)$ is a Lie group endowed with a bi-invariant metric and $V_k(\mathbb{C}^n)$ is a homogeneous space, which fits into the setting of one of the examples proposed by O'Neill in "The fundamental equations of a submersion", where he claims that such submersions have totally geodesic fibres.
Nevertheless, I have not found any reference discussing whether $\pi_2$ has totally geodesic fibres or not.