Suppose someone claims that all morality is relative, but when pressed on whether this would apply even to murder, they act evasive and refuse to give a clear answer. A critic might conclude that this person is disingenuous in refusing to accept the clear logical consequences of their belief.
However, imagine that there's a really strong social stigma against asserting that murder might not be bad, to the point of permanently damaging such a person's reputation, even though there's no consequence for making the actually stronger claim that all morality is relative. The relativist might therefore see the critic as the one who is disingenuous; trying to leverage social pressure against them instead of arguing on the basis of reason.
Thus in the right circumstances, each side can quite reasonably see the other as disingenuous. I suspect that everyone will have experienced both sides of the coin at different times depending on the issue being discussed.
This post was produced with the support of the EA Hotel