Oh I am wrong here; the reduction doesn’t work.
I don’t see how to construct an example though.

Doesn’t that make all the A_i equivalent?

So for every production B \to \alpha we have m productions of the form A_i \to \alpha.

I’m not sure how the suggestion goes. If G has an SPP whose B-component is a union of m sets of the form C_i^{\triangleleft} \cap D_i^{\overline{\triangleleft}}, are we supposed to replace B with m new nonterminals A_1, \dots, A_m so that a new SPP will have the set C_i^{\triangleleft} \cap D_i^{\overline{\triangleleft}} as its A_i-component? What will the productions be that have A_i as their left-hand side?

My original intuition was that of course more context-free languages are included because FC_L(k,l,m) contains many more sets than FC_L(k,l). But it’s very difficult to give examples of CFLs that differentiate the two. Perhaps you don’t get any more CFLs.

Sound pre-fixed points work for both approaches. For the original primal approach, an SPP must consist of the closures of finite sets of strings. To generalize, you could think of changing either “closure” or “sets of strings”. There is some discussion in the concluding section of our paper. I can probably give more details during the Q/A period.