Achievable Cantorvals almost without reversed Kakeya conditions

Examples of achievable Cantorvals are constructed with reversed Kakeya conditions only on a set of asymptotic density zero which answers in positive the Problem 5.2 from Marchwicki and Miska (2021). Additionally, the Lebesgue measure of the boundaries of these Cantorvals is found to be zero which does not answer the still open problem of existence of achievable Cantorvals with boundaries of positive measure.