Claims

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1968_01_01_schmidt: Schmidt's 1968 theorem that for every sequence in the unit interval some interval has unbounded counting discrepancy, refereed in the Quarterly Journal of Mathematics; it answers the question yes.

1972_01_01_schmidt: Schmidt's 1972 theorem that for every sequence in the unit interval the anchors $x$ at which $D_N([0,x))$ stays bounded are at most countable, so all but countably many anchored intervals answer the question yes; refereed.

1980_12_01_tijdeman_wagner: Tijdeman and Wagner's 1980 theorem that every sequence in $[0,1)$ has $\limsup_N\lvert D_N([0,x))\rvert/\log N\ge1/400$ for almost all $x$, so almost every anchored interval answers the question yes; refereed.

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