Is it more important for a leader to publicly show support during conflicts, or to strategically remain quiet?

I believe that it all depends on how the leader plans to lead his/her people. Some leaders may make so much noise publicly and that's all some are just all talk and no action while others do the actions and let it speak for them. Leaders are more important depending on the audience,topic and way of life.

it is probably more important for a leader to show support during a conflict.

It depends, there are situations like Taiwan where it's very grey and even the country itself doesn't really hunger for true independence so keeping quiet and just letting things be is a very strategically sound move but there are also times like Ukraine where it's very obvious who is in the wrong and there needs to be an international outcry to set things right for everyone involved.

If they are a leader, they should publicly show support during conflicts.