Strong convergence of projections in $B(H)$

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)

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Let $p_k=\frac12\,(e_{11}+e_{1,k+1}+e_{k+1,1}+e_{k+1,k+1})$. Then $q_kp_kq_k=\frac12\,e_{11}$ for all $k$, and the sequence $\{p_k\}$ converges weakly to $\frac12\,e_{11}$. As the limit is not a projection while every $p_k$ is, we conclude that the sequence does not converge strongly.

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