 Dependent uniform sampling variance lower bound

Let $Z_1,\dots ,Z_n$ each be uniform on $[0,1]$, with arbitrary dependence. For centered unit-norm $f\in L^2[0,1]$, putThenPartition $[0,1]$ into $k$ equal intervals and test normalized differences of two interval indicators. Averaging over the two intervals reduces the claim to ${\sum }_p{N}_p^2\ge {\sum }_p{N}_p=n$ for their occupancy counts, and then let $k\to \infty$.