In bosonic string theory the Polyakov action can be put in into conformal gauge. It is then possible to show that the resulting gauge fixed action is conformally invariant. Actually it's shown that it's invariant under the combined actions of conformal transformations and Weyl transformations, but it's called 'conformal invariance'.
Since this invariance applies to the gauge fixed action, does this mean that the pre-gauge-fixed Polyakov action is conformally invariant also? It's just easiest to demonstrate in conformal gauge since the metric is flat and the determinant of the metric becomes unity?
In general I would expect that all symmetries of a gauge fixed action are also symmetries of the pre-gauge fixed action. On the other hand not all symmetries of the pre-gauge-fixed action are symmetries of the gauge fixed action. This is because the process of gauge fixing means fixing values that would otherwise freely vary according to those symmetries that are gauge fixed.
Is the above reasoning correct? Can we say the pre-gauge-fixed Polyakov action is conformally invariant?