Separation conjecture for core and non-core rules in the calculus of structures

Let $T$ and $R$ be structures, and let $\mathsf{SKS}$ be the calculus-of-structures system for classical propositional logic. Partition its rules into the core fragment and the non-core fragment, with identity and cut written separately. Let $T',T”,R”,R'$ be intermediate structures.

Separation conjecture. For every derivation $T\vdash_{\mathsf{SKS}}R$, there is a derivation of the form

$$T\vdash_{\mathrm{non-core}}T'\vdash_{\{\text{identity}\}}T”\vdash_{\mathrm{core}}R”\vdash_{\{\text{dual identity}\}}R'\vdash_{\mathrm{non-core}}R.$$

The conjecture asks whether switch and medial, and more generally the core and non-core fragments, can be separated into phases. The source presents this as a problem for future work and notes that related separation results had been proved for two other systems.