The countable width-two lattice decomposition conjecture

Let $L$ be a countable lattice of width two satisfying Whitman's condition, and let $N$ be a chain order-isomorphic to $\omega$, its dual, or the chain

$$\dots < -2 < -1 < 0 < 1 < 2 < \dots.$$

For each $i\in N$, let $P_i$ be the disjoint union of two chains $A$ and $B$ and the set $\{0_i,1_i\}$, such that for every $a\in A$ and $b\in B$, $a+b=1_i$ and $ab=0_i$.

The countable width-two lattice decomposition conjecture. The lattice $L$ is semidistributive and

$$L=\bigcup_{i\in N}P_i.$$

Moreover, $P_i\cap P_j=\{0_j,1_i\}$ if $i\prec j$ in $N$, and $P_i\cap P_j=\varnothing$ otherwise.

This conjecture proposes an explicit description of countable width-two lattices satisfying Whitman's condition; the source suggests that it may follow from a modification of the preceding arguments and may be easier to prove than Jónsson's conjecture. Its resolution is not supplied here.