Suppose that you're given a non-perturbative $S$-matrix that corresponds to some Wightmanian QFT. By this I mean that you're given a Hilbert space and a unitary operator $S$ that acts on the Hilbert space, and transforms appropriately under the Poincare symmetries.

Is it possible to reconstruct the QFT from its $S$-matrix? By reconstructing the QFT I mean obtaining the Wightman functions, or equivalently, the operator distributions corresponding to the quantum fields.

I'm not asking about a general reconstruction algorithm – I'm aware that it isn't known presently. I'm asking about the possibility of such  reconstruction in general. For example, if there is a counterexample of two different Wightmanian QFTs having the same $S$-matrix, that would answer my question in the negative. It is frequently stated that onshell amplitudes contain less information than offshell correlation functions, it would be nice to have a concrete example demonstrating this.