The question actually is limited to a very specific case.
The following takes place in a fixed Hilbert space.
Let $(p_i)_i, (q_i)_i, p, q$ projections (resp. nets of projections) so that $\underbrace{p_i\searrow p}_{(p_i)_i \textrm{mon. decr.}}$ and $\underbrace{q_i \longrightarrow q}_{_{SOT}}$.
Question. If $(\forall{i})\ p_i \wedge q_i=0$, does it also hold, that $p\wedge q=0$?