Framing-lattice invariance of linear-interval enumerators

Let $G$ be a graph, and let $F_1$ and $F_2$ be framings of $G$. For each $k=0,1,2,\ldots$, consider the linear intervals of length $k$ in the framing lattices $\mathscr{L}_{G,F_1}$ and $\mathscr{L}_{G,F_2}$. Framing-lattice linear-interval conjecture. The lattices $\mathscr{L}_{G,F_1}$ and $\mathscr{L}_{G,F_2}$ have the same number of linear intervals of length $k$ for every $k=0,1,2,\ldots$. This is a purely enumerative conjecture supported by extensive computational evidence; its general validity remains open.