The strengthened generating-function conjecture for $\textbackslash \mathcal{P}$-Knuth graphs

Let $\textbackslash \mathcal{P}$ be a unit interval order, and let $\textbackslash \Gamma_V$ be a connected $\textbackslash \mathcal{P}$-Knuth equivalence graph with generating function $\textbackslash \gamma_V$. For a $\textbackslash \mathcal{P}$-tableau of shape $\textbackslash \lambda$, write its reading word for the corresponding permutation. The theorem referenced in the claim is the stated generating-function formula for natural unit interval orders avoiding $\textbackslash \mathcal{P}_{(3,1,1),5}$ and $\textbackslash \mathcal{P}_{(4,2,1,1),6}$.

Main conjecture. The claim of the stated theorem is true for all unit interval orders $\textbackslash \mathcal{P}$, except for its final assertion concerning equality of tableau-shape lengths and genuine $\textbackslash \mathcal{P}$-height.

The claim extends the paper's proved theorem beyond the two avoidance hypotheses. The supplied text does not provide a proof or resolution of this assertion, so it remains open.