# Can relativization technique be applied to natural NP-complete languages?

Levin [1] defined distNP is the distributional problem (L,D), where L ∈ NP, and D is an ensemble of efficiently samplable distributions over problem instances. We say that a distNP problem (L,D) is hard if there is no efficient algorithm that solves the problem (with high probability) on instances sampled from D.

Holenstein and Kunzler [2]asked the following question: Question: Does $$NP\nsubseteq BPP$$ imply that there exists a hard problem in $$distNP$$?

Impagliazzo [3] showed that any proof that gives a positive answer to the above question must use non-relativizing techniques. More precisely, it is shown that there exists an oracle $$O$$ such that $$NP^O\nsubseteq BPP^O$$, and there is no hard problem in $$distNP^O$$.

My question: Does there exists $$X\subset NP$$ such that there exists an oracle $$O$$ such that $$X^O\nsubseteq BPP^O$$, and there is no hard problem in $$distX^O$$? More specifically, how about if $$X$$ is a natural NP-complete problem which is can be represented by paddable language (i.e. we can encode arbitrary additional information, without changing the membership of the instance in the language.)?