Before revision:  This  property is always satisfied if the test is based on a polyhedral risk measure such as Expected Shortfall, but it generally fails , even in a convex world, if we depart from polyhedrality .
After revision:  This "stability" property is always satisfied if the test is based on a polyhedral risk measure such as Expected Shortfall, but it generally fails , even in a convex world, if we depart from polyhedrality .