Converse transfer of ultrafilter orders from a chainable continuum to its inverse-limit model

Determine whether, for every chainable continuum X that is homeomorphic to an inverse limit of arcs Y = lim←(I_i,f_i)_{i=1}^∞, every ultrafilter order ≤_U^D on X (defined via a sequence of chains D covering X with mesh(D_n) → 0 and a non-principal ultrafilter U on ℕ) arises from some ultrafilter order ≤_U^{(I_i,f_i)_{i=1}^∞} on Y in the sense that there exists a homeomorphism h': X → Y with x ≤_U^D y if and only if h'(x) ≤_U^{(I_i,f_i)_{i=1}^∞} h'(y).

Background

The paper introduces two constructions of ultrafilter orders on chainable continua: one using sequences of increasingly fine chains (Definition 3.1) and another using inverse limits of arcs (Definition 3.7). Theorem 3 (Transfer) shows that any ultrafilter order defined on the inverse limit Y induces an ultrafilter order on the homeomorphic chainable continuum X.

The authors explicitly note that the reverse direction is unknown: whether any ultrafilter order defined via chains on X can be represented as one coming from the inverse-limit construction on Y under a homeomorphism, thus making the two approaches fully equivalent.