Knutson's puzzle conjecture for two-step flag varieties

Let $F(a,b;N)$ be a two-step flag variety, and let $X_u$, $X_v$, and $X_w$ be Schubert varieties indexed by permutations $u,v,w\in S_N$. Write $J(u)$, $J(v)$, and $J(w)$ for their associated $012$-strings. A puzzle is a tiling by the six specified $012$-puzzle pieces, allowing rotations but not reflections.

Knutson's conjecture. The integral

$$\int_{F(a,b;N)} [X_u]\cdot [X_v]\cdot [X_w]$$

is equal to the number of puzzles whose north-west, north-east, and south boundary labels, read clockwise, are $J(u)$, $J(v)$, and $J(w)$, respectively.

This is a proposed combinatorial rule for Schubert structure constants on two-step flag varieties. It had been verified computationally for all such varieties with $N\leqslant 16$, while the broader conjecture for all partial flag varieties was disproved by counterexamples.