Tableaux problem 44

44.[Stan1990a]*sec. 2 For any vector $\mathbf{r}$ there is at most one $\mathbf{r}$-differential distributive lattice $L(\mathbf{r})$. If we consider the class of $\mathbf{r}$-vectors that are arithmetic progressions with a given difference $\Delta$, then the class $L(\mathbf{r})$ of differential lattices that are differential posets with such $\mathbf{r}$-vectors is closed under cartesian products. Conversely, if the cartesian product of two sequentially differential posets each with at least two ranks is a sequentially differential poset, then both the factors and the product have $\mathbf{r}$-vectors that are arithmetic progressions and they all have the same difference $\Delta$. Applying [Stan2012a]*Exer. 3.51(a) soln. shows that $\Delta\leq 0$. If $\Delta=0$, they are ordinary differential posets. But if $\Delta<0$, they must be finite. Do there exist any finite distributive differential posts with $\mathbf{r}$-vectors that are arithmetic progressions other than the cartesian exponentials of the 2- and 3-element chains?[Stan1990a]*Exam. 2.5 Can the cases with $\Delta<0$ be fully classified?