Random independent misclassification error is an inappropriate noise model for a membership query (MQ) oracle because for any noise rate $\eta<1/2$ one can eliminate noise to an arbitrary extent by simply querying the oracle $p(1/(1-2\eta))$ times with the same input for some polynomial $p(\cdot)$ and taking the majority label returned.
The random persistent misclassification noise model of Goldman, Kearns, and Schapire, plugs this "loophole" by stipulating that an oracle that assigns a label $l$ to some sample $x$, always assigns label $l$ to sample $x$.
My question is: What are the known best upper bounds on the time and queries required to solve the learning parities problem with membership queries and random persistent misclassification noise for some constant value of $k$ (the number of relevant attributes)? Is the problem known to be solvable in $O(n \log (n/\delta))$ time and $O(\log (n/\delta))$ queries w.r.t. $n$, the number of attributes and $\delta$, the probability of error?