Equality of 𝔈(b_r G) and e_{c_m X}G for ultrahomogeneous LOTS

Determine whether the equality 𝔈(b_r G) = e_{c_m X}G holds for G = Aut(X) with the topology of pointwise convergence τ_p acting on an ultrahomogeneous linearly ordered topological space X, where b_r G is the Roelcke compactification of G and c_m X is the least linearly ordered compactification of X.

Background

Theorem ‘Roelcke precomp4-3-1’ establishes that for an ultrahomogeneous LOTS X, the Ellis compactification e_{c_m X}G is the completion of (G, R_{Σ{c_m X}}), that G is Roelcke precompact, and that b_r G < e{c_m X}G. The map 𝔈 is defined earlier as sending a G-compactification of G to the corresponding Ellis compactification.

The authors ask if applying 𝔈 to the Roelcke compactification b_r G yields precisely the Ellis compactification e_{c_m X}G obtained from the maximal linearly ordered compactification c_m X. Resolving this would clarify the precise relationship between the Roelcke and Ellis compactifications in the ultrahomogeneous LOTS setting.