Action of a projection on a sequence of pairwise orthogonal subspaces. Let us consider a sequence of finite-dimensional subspaces $\{H_n\}$ with $dim H_n=dim H_m$  and $H_n\perp H_m$ ($m\neq n$) in the Hilbert space $H$. Does there exist any projection $P$ satisfying in the following relations: 
$$0\neq \overline{P(H_n)}=\overline{P(H_m)}=\overline{P(\cup_{j\geq1} H_j)}$$
where $m\neq n$.
My answer was wrong. The mistake that invalidates the whole argument happened between $(5) $ and $(6)$, where 
$$
\lambda_{jk}P_{jj}=\lambda_{jk}\lambda_{kj}P_{kk}
$$
should have been
$$
\lambda_{jk}P_{jj}=\lambda_{jk}\lambda_{kj}^2P_{kk},
$$ which gives no new information whatsoever.
Have you tried anything?

Try the simple case $H = \mathbb{C}^n$ and $H_i = \{(0,0, \dots, 0, z, 0, \dots) : z \in \mathbb{C}\}$ is the span of the $i$th standard basis vector. (Try $n = 2$ and $n = 3$.)

Can you find a projection with the above properties in this case? If not, then why not?

Can you extend this to the general case?