Greatest-element conjecture for ribbons with fixed dimensions

Consider the partial order on $G$-equivalence classes of skew shapes defined by $A\leq B$ when $a_{A,\nu}\leq a_{B,\nu}$ for every partition $\nu$, equivalently when $G_B-G_A$ is $G$ positive. Ribbons with a fixed number of rows and columns form a connected component of this poset. Greatest-element conjecture. In addition to its least element, this set of ribbons has a greatest element. The source gives no proof or resolution of this assertion.