Turning off specularity is rarely going to be appropriate because in real life even things we consider "not shiny" are still doing some shine.

Here I modulated the blue box so that the window panes are still highly specular, but the blue plastic is not. Yet it has a realistic non-zero amount of shine. It's just very low and very spread out.

If you took that away completely, it would not look real.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)