On duality gap in linear conic problems

In their paper ‘Duality of linear conic problems’ Shapiro and Nemirovski considered two possible properties (A) and (B) for dual linear conic problems (P) and (D). The property (A) is ‘If either (P) or (D) is feasible, then there is no duality gap between (P) and (D)’, while property (B) is ‘If both (P) and (D) are feasible, then there is no duality gap between (P) and (D) and the optimal values val(P) and val(D) are finite’. They showed that (A) holds if and only if the cone K is polyhedral, and gave some partial results related to (B). Later Shapiro conjectured that (B) holds if and only if all the nontrivial faces of the cone K are polyhedral. In this note we mainly prove that both the ‘if’ and ‘only if’ parts of this conjecture are not true by providing examples of closed convex cone in ${ℝ}^4$ for which the corresponding implications are not valid. Moreover, we give alternative proofs for the results related to (B) established by Shapiro and Nemirovski.