genetics - Is the variance of a trait related to its rate of evolution?

Posed with a question that has me stuck:

Is increased variance of a phenotypic trait in a population associated with slower evolution of that trait?

I have to either 'disagree or agree' and argue my point. And I disagree. For this question, I think that if the variance of phenotypic trait has a genetic basis (additive genetic variance =/= 0) then increased additive genetic variance should speed up it's evolution. The greater the additive genetic variance, the greater the response to selection in the next generation. So increased additive genetic variance should increase the rate of evolution of that trait.

However, at the same time, if there is variation in a phenotypic trait but it does not have a genetic basis (narrow sense heritability is zero) and the basis for variation is due to an environmental, dominance, or interactive effect, then increased variation would have no effect in the rate of evolution. Or so I think. I'm not too sure how the rate of evolution would be affected in the case of a non-genetic basis of variation.