We consider ribbon shapes, not necessarily connected, whose rows, with at least two boxes in each, are in monotone length order. These ribbons are uniquely defined by a pair of partitions: the row partition consisting of the row lengths in decreasing order, and the overlapping partition whose entries count the total number of columns with two boxes in the successive ribbon shapes obtained by sequentially subtracting the longest row. The support of such ribbon Schur functions, considered as a subposet of the dominance order lattice on partitions, has the row partition as bottom element, and, as top element, the partition whose two parts consist of the total number of columns, and the total number of columns of length two respectively. We give a complete system of linear inequalities in terms of the partition pair defining the aforesaid ribbon shape under which the ribbon Schur function attains all the Schur interval when expanded in the basis of Schur functions. We then conclude that the Gaetz-Hardt-Sridhar necessary condition for a connected ribbon to have full equivalence class is equivalent to the condition for a monotone connected ribbon to have full Schur support. That is, the set of partitions with full equivalence class is a subset of those monotone connected ribbons with full Schur support. M. Gaetz, W. Hardt and S. Sridhar conjectured that the necessary condition is also sufficient which translates now to every monotone connected ribbon with full Schur support has full equivalence class. The main tool of our analysis is the structure of the companion tableau of a ribbon Littlewood-Richardson (LR) tableau detected by the descent set defined by the composition whose parts are the ribbon row lengths.