Non trivial lower bound for $\sup_P \|Px\|$ where $x$ is fixed, and the supremum if over all full-rank projection matrices $P \in R^{k \times n}$

Question

Let $1 \le k \le n$ be integers and let $G_{n,k}$ be the grassmannian of $k$-dimensional subspaces of $\mathbb R^n$. For any $U \in G_{n,k}$, let $P_U$ be the orthogonal projection onto from $\mathbb R^n$ to $U$. Fix a unit vector $x \in \mathbb R^d$; for example, WLOG, take $x=(1,0,\ldots,0)$.

Question. What is a good lower-bound for $\alpha_{n,k} := \sup_{U \in G_{n,k}}\|P_U x\|$ ?

Notes

• Alternatively, one can rewrite $\alpha_{n,k} = \sup_P \|Px\|$, where the supremum if over all rank $k$ projection matrices.

• I know that the answer has to be (much ?) larger than $c\sqrt{k/n}$, since this is the naive lower-bound one gets by considering random $U$ and taking expectations (w.r.t Haar measure on $G_{n,k}$) instead of supremum.