# Complete problems for $(NP\cap CoNP)/poly$ class and universal representations

It is conjectured that $$NP\cap CoNP$$ does not have a complete problem with respect to the polynomial-time many-to-one reductions. I would like to know the current knowledge about the nonuniform versions of this question:

Q1. Does $$(NP\cap CoNP)/poly$$ have a complete problem with respect to the $$P/poly$$ many-to-one reductions?

A related question is about the existence of a universal representation of $$(NTIME(n^a)\cap CoNTIME(n^a))/n^b.$$ where $$a$$ and $$b$$ are arbitrary fixed constants. By a universal representation, I mean the existence of a function $$f(e,x)\in (NTIME(n^c)\cap CoNTIME(n^c))/n^d$$ for some fixed $$c$$ and $$d$$ ($$|e|=|x|^w$$ for a fixed $$w$$) such that for every $$g(x)\in (NTIME(n^a)\cap CoNTIME(n^a))/n^b,$$ we have $$\forall n\in\mathbb{N},\exists e_n\in\{0,1\}^{n^w},\forall x\in\{0,1\}^n\: f(e_n,x)=g(x).$$

Does such a representation exist?