According to a new paper, the answer is no:
For three academic years (2011-2014), the Harvard Law School’s Shareholder Rights Project (SRP) operated a clinical program assisting institutional investors on board declassification proposals. This paper analyzes the SRP as a quasi-natural experiment to examine the value implications of classified boards. Consistent with the SRP causing exogenous declassifications, SRP targets that declassified their boards had greater ex-ante value and profitability than non-targeted companies declassifying in the same period. Declassifying SRP targets, and especially targets more engaged in research and innovation, declined in firm value after declassification, which wealth effect is directly attributable to declassification rather than other activism-related channels.
Cremers, Martijn and Sepe, Simone M., Board Declassification Activism: The Financial Value of the Shareholder Rights Project (April 2, 2017). Available at SSRN: https://ssrn.com/abstract=2962162
Lucian Bebchuk replies here.