Can we say that the curl of $E$ for an electromagnetic wave is zero?

here (page 11) the following statement is written about the curl of the electric field in a TEM wave on a transmission line (⊥ indicates the component on the plane orthogonal to the propagation axis z):

Since $B_z = 0$, Faraday’s law tells us that,

$0 = −frac{∂B_z}{∂t} = (∇ × E)_z = ∇_⊥ × E_⊥ e^{j(ωt−kz)}
, $

where $∇_⊥ = (∂/∂x, ∂/∂y)$. Hence, the 2-dimensional, time-independent
field $E_⊥$ has zero curl, and so can be deduced from a (static)
2-dimensional scalar potential $V_⊥$ according to,

$E_⊥ = −∇_⊥V_⊥$.

Can we apply the same logic on an ordinary (TEM) electromagnetic wave? It seems quite strange to me, because zero curl of E means something like an electrostatic situation, and so how can the wave propagate in such a situation? But at the same time, I cannot find the mistake in such a proof if applied to an ordinary electromagnetic wave.

My question in other words: is it possible to say that $E$ may be expressed as the transverse (2D) curl of an electric potential $V$ in an electromagnetic wave?