Suppose that $U$ and $W$ are subspaces of $\mathbb{R}^n$ such that $U \subseteq W$, with respective dimensions $m = \dim(U)$, $s=\dim(W)$. Denote the (perpendicular) projection matrices onto $U$ and $W$ as ${\bf P}_U$ and ${\bf P}_W$, respectively. Let us also define the orthogonal complement $W^\perp$, with projection matrix ${\bf P}_W^\perp = {\bf I} - {\bf P}_W$.
Then the subspace $Z = W \cap U^\perp$ has projection matrix ${\bf P}_Z = {\bf P}_W - {\bf P}_U$.
Let ${\bf x} \in \mathbb{R}^n$ be normally distributed with zero mean and covariance ${\bf I}_n$. One has $$ \|{\bf x} \|^2 = \|{\bf P}_U {\bf x} \|^2 + \|{\bf P}_Z{\bf x} \|^2 + \|{\bf P}_W^\perp{\bf x} \|^2,$$ and each of the three terms in the right-hand side is chi-square distributed with $m$, $s-m$ and $n-s$ degrees of freedom, respectively; moreover, they are statistically independent (as per Cochran's Theorem). Intuitively, this is due to the fact that the subspaces $U$, $Z$ and $W^\perp$ are pairwise orthogonal.
My question is the following. Consider the random variables $$ r_1 = \frac{ \|{\bf P}_W{\bf x} \|^2}{\|{\bf P}_W^\perp {\bf x} \|^2} \sim F_{s,n-s} \qquad \mbox{and} \qquad r_2 = \frac{ \|{\bf P}_U{\bf x} \|^2}{\|{\bf P}_Z{\bf x} \|^2} \sim F_{m,s-m}. $$ Are they statistically independent? Empirical evidence is supportive, and I conjecture that they are, but I have been unable to obtain a formal proof. Note that $r_1$ is the ratio of the energy of $\bf x$ in $W$ to that in $W^\perp$ (its orthogonal complement in $\mathbb{R}^n$), whereas $r_2$ is the ratio of the energy of $\bf x$ in $U\subseteq W$ to that in $Z$, which is its orthogonal complement in $W$.
I think the question is of independent interest, but in any case, I arrived at this from a hypothesis testing problemin which, on one hand, I test $H_0: {\bf x} \in W$ vs. $H_1: {\bf x} \notin W$ (with test statistic $r_1$) and on the other, $H_0': {\bf P}_W{\bf x} \in U$ vs. $H_1': {\bf P}_W{\bf x} \notin U$ (with test statistic $r_2$). Thanks in advance!