  The recent study by Waclawczyk et al. [Phys. Rev. Fluids 6, 084610 (2021)] on
conformal invariance in 2D turbulence is misleading as it makes three incorrect
claims that form the core of their work. We will correct these claims and put
them into the right perspective: First, the conformal invariance as proposed by
Waclawczyk et al. is not related to the result that zero-isolines of the scalar
field in the inverse energy cascade display a Schramm-Loewner evolution (SLE).
Second, the conformal invariance is not a Lie-group for all values of the
scalar field since it inherently violates the smoothness axiom of a Lie-group
action, with the effect that a physical PDF gets mapped to a non-physical one.
Third, although Waclawczyk et al. recognize that their conformal invariance
does not constitute a symmetry but only a weaker equivalence transformation, it
is still not classified correctly. The claim that their equivalence can map
between solutions is not true. This fact will be demonstrated by using an
illustrative example.
