Equality of C and C′ under coflatness and flatness on the same side

Determine whether, for a Hopf algebra H with bijective antipode, if H is left coflat over a left H-module factor coalgebra C and left flat over the associated right coideal subalgebra A = coCH, then C necessarily equals C′ = H/HA+; equivalently, prove or disprove that the canonical comparison C′ → C is an isomorphism under these hypotheses.

Background

Theorem 7.4 shows that when H is left coflat over C and right flat over A = coCH, H becomes left faithfully coflat over C, leading to C = H/HA+. However, when both coflatness over C and flatness over A hold on the same (left) side, the authors cannot conclude that C equals C′ = H/HA+.

Later, Section 10 revisits this via colocalization: under left coflatness over C and left flatness over A, the canonical surjection C′→C exhibits C′ as a perfect right colocalization of C, and the authors note no known examples where C′≠C, underscoring the open status of the equality question.