Equivalence and abnormal-Carathéodory-equivalence of optimal control problems

In the terminology of Constantin Carathéodory, two problems are Carathéodory-equivalent when their respective Lagrangians differ by a total derivative; the corresponding extremals are then related. The source further considers the analogous notion for abnormal extremals, called abnormal-Carathéodory-equivalence. Abnormal-Carathéodory equivalence conjecture. Two problems of optimal control are equivalent if, and only if, they are abnormal-Carathéodory-equivalent. The claim would identify ordinary equivalence with the correspondence of abnormal extremals and is motivated by the observation that Carathéodory-equivalent problems have the same conservation laws; its status is not resolved in the supplied source.