The surge of interest in socially responsible investment (SRI) over the last decade generates a shift in investors' belief but also new challenges to assess. Both, passive and active investment management step in this new field, integrating extra financial data within investment process. However, this trend opens up a Pandora's box for active portfolio managers. The ESG-related track error induced by the integration non-pecuniary factors within the decision-making process emerges. This paper investigates whether investors may favor securities presenting features in line with fundamental portfolio guidelines, namely better ESG quality and optimal Index tracking. Using US-listed firms of the USA MSCI Index and Refinitiv ESG score from 2013 to 2021, we propose a double sort methodology to assess our assumption. We found that Low ESG-Beta stocks are relinquished by investors in order to answer new constrains and generate higher risk adjusted returns. However, findings do not confirm if this phenomena happens in profit of High-ESG-Beta securities. Our results also involve that this possible phenomena is relatively recent undertaking.