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on Tableaux problem 44

The statement

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?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.