For two closed operators $A,B$, can we say that $P_i\uparrow 1$ and $(A-B)P_i =0$ implies $A-B$?

Question

Let $A,B$ be two closed operators on a Hilbert space. Let $P_i$ be an increasing net of projections with $P_i\uparrow {\bf 1}$. If $(A-B)P_i =0$ for every $i$ (if necessary, we may assume that $AP_i,BP_i$ are bounded and self-adjoint for every $i$), do we have $A-B=0$?

Answer 1

Multiplication is a separately continuous map in WOT, SOT and norm.

I would guess the same is true for any reasonable locally convex topology in B(H).