The product-of-chains conjecture for distributive cross section lattices

Let $\Lambda$ be a distributive cross section lattice of the $J$-irreducible monoid of the pair $(G_0,\rho)$, where $G_0$ is a simple algebraic group of one of the types $A_{n-1}$, $B_{n-1}$ or $C_{n-1}$. Let $J_0$ and $\Delta=\{\alpha_1,\ldots,\alpha_{n-1}\}$ be the set of simple roots as in the preceding theorem. Suppose that $J_0$ has connected components $J_0^{(1)}=\{\alpha_1,\alpha_2,\ldots,\alpha_k\}$ and $J_0^{(2)}=\{\alpha_l,\alpha_{l+1},\ldots,\alpha_{n-1}\}$, with the possibility that $J_0^{(1)}=\emptyset$ or $J_0^{(2)}=\emptyset$. Product-of-chains conjecture. Then $\Lambda^*$ is isomorphic to the product of chains

$$C_{k+2}\times C_{n-l+2}\times C_2^{l-k-3}=C_{k+2}\times C_{n-l+2}\times C_2\times\cdots\times C_2.$$

The claim gives a precise structure for distributive cross section lattices in the indicated classical types; the supplied text presents it as motivated by the rational smoothness discussion and gives no evidence that it has been resolved.