Sustainable living has emerged as the need of the hour for mankind in present times. Practitioners, as well as scholarship in the area, are divided over the comparison of financial returns from sustainable indexes vis-à-vis conventional indexes, causing investors' dilemma. These questions loom larger during the times of global crises, such as COVID-19, which have brought sustainability concerns to the limelight. This dilemma of the investors leads us to approach the study on hand. We study the Thomson Reuters/S-Network global indexes (as a proxy for sustainability-based indexes), and their corresponding alternatives, using the daily closing prices from 1^st January 2011 to 29^th June 2020. We apply the time-frequency-based Granger-Causality test, and further attempt to understand the coherence between these indexes before and during the COVID-19 period by using the Wavelet Coherence and phase-difference mechanisms. Our results suggest short-run uni-directional causality from sustainable indexes to conventional indexes whereas bi-directional causality in medium and the long-runs. The coherence is particularly stronger at low frequencies, indicating the long-run coherence with sustainable indexes in the lead during COVID-19. The results and conclusions of the study have important implications for different audiences. The portfolio and fund managers can prefer to invest in such markets to avail of higher returns over a longer period.