A magic unitary is a unitary matrix $u=(p_{ij})_{ij}$ whose entries are all projections (in some Hilbert space) and in each row they sum to the identity and same holds for each column $(\sum_i p_{ij}=1=\sum_j p_{ij})$. My question is, if there is a sequence of "almost magic unitaries", by which I mean matrices $u_k=(q_{ij}^k)$ whose entries are all projections BUT they only sum to identity in each row and NOT in each column but close to identity ($\sum_j q_{ij}^k=1$ and for each j, $\sum_i ||q_{ij}^k-1||_2\rightarrow 0$) such that $u_k\rightarrow u$ in $||.||_2$ norm, Can one find actual magic unitaries that are close to this "almost magic unitaries" which approximate the original magic unitary?

It's similar to finding an orthogonal set of projections summing to identity from a set of almost orthogonal positive operators almost adding up to the identity.