Are unitary conformal field theories with central charge c>=1 realizable on the lattice?

As John Cardy showed in 1986 in "Operator content of two-dimensional conformally invariant theories", a conformal field theory (CFT) which is both unitary and has a finite number of primary operators must have a central charge $$c<1$$.

There are a number of critical lattice models that have a CFT continuum limit, e.g. the Ising and Potts model that both have $$c<1$$. It should be impossible for a lattice model to represent an infinite number of (lattice representations of) primary operators, so does Cardy's result rule out any lattice model with unitary time evolution that has a $$c \geq 1$$ CFT continuum limit? The only possible loophole to this argument seems be a lattice model that represents a divergent number of primary operators in the continuum limit, does such a model exist?