*(I asked this question at math stackexchange 4 months ago, but received no answers)*

Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by $$ q_k=\sum_{n=1}^ke_{nn}. $$ Let $\{p_1,p_2,\ldots\}\subset B(H)$ be a sequence of orthogonal projections in $B(H)$ with the property that $q_kp_hq_k=q_kp_kq_k$ whenever $h\geq k$ (i.e. the sequence "fixes" the upper left corner as the index grows).

Question:Does the sequence $\{p_k\}$ converge strongly?

*(my gut feeling is that it should, but after a while thinking about it I couldn't get neither a proof nor a counterexample; it is easy to show that the sequence converges weakly so it would be enough to prove that the limit is a projection, but I got nowhere through this route either)*