Before revision:  We answer this question in the affirmative by designing a computationally efficient algorithm with near-optimal label complexity  O\big( s \log^3 d \cdot \log ^4\frac{1{\epsilon}} {\epsilon}}} \big) and noise tolerance \eta = \Omega(\epsilon), where \epsilon \in (0, 1) is the target error rate.
After revision:  We answer this question in the affirmative by designing a computationally efficient algorithm with near-optimal label complexity O\big( s \log^3 d \cdot \log ^4\frac{1{\epsilon}} ^3\frac{1{\epsilon}}} \big) and noise tolerance \eta = \Omega(\epsilon), where \epsilon \in (0, 1) is the target error rate.