The mortar-and-brick conjecture for $K_2$-QPDSs in the triangular lattice

Let $\Delta$ be the triangular lattice. A $K_2$-QPDS $S$ in $\Delta$ has an associated hexagonal structure ${\mathcal P}(S)$ whose hexagons have types $1$, $2$, and $3$. Mortar-and-brick conjecture. No $K_2$-QPDS in $\Delta$ contains two different triples of pairwise adjacent hexagons in ${\mathcal P}(S)$ with pairwise different types $1$, $2$, and $3$. This conjecture concerns the apparent restriction on viewing the triangles of ${\mathcal P}(S)$ as mortar binding and separating its hexagons as bricks; the source does not indicate that it has been resolved.