Can you please refer any paper or forward your ideas that what are the challenges could evolve while operationalizing the equity and justice dimensions for sustainability?
I would recommend looking at the detailed description of the UN Sustainable Development Goals (SDGs) if you haven't already. https://sustainabledevelopment.un.org/topics/sustainabledevelopmentgoals
They provide targets and guidelines in response to 17 sustainability challenges. Many people will recognise the pictures as they are popping everywhere, but few actually know the details for each goal. For equity and justice, you can read the targets under the 10th and 16th SDGs: 10. Reduced inequalities, and 16. Peace and justice.