Finding a certificate if E=NE If $\textbf{P}= \textbf{NP}$ then for every language from $\textbf{NP}$ there exists an algorithm of finding a certificate in polynomial time. 

Assume that $\textbf{E} = \textbf{NE}$. Is it true that there is an algorithm of finding a certificate in exponential time for every language from $\textbf{NE}$?
R. Impagliazzo ; G. Tardos shown that there exists  an oracle  relative to which
$\textbf{E} = \textbf{NE}$ , yet there is an exponential-type search problem
that requires doubly exponential randomized time.

The similar situation with collapsing of exponential hierarchy.