Quadratic generation conjecture for join-meet binomial algebras of simple planar distributive lattices

Let $L$ be a simple planar distributive lattice, and let ${\mathcal R}_K(L)$ be its join-meet binomial algebra with defining ideal $I_{{\mathcal R}_K(L)}$. An interval is a sublattice of $L$ consisting of all elements between two fixed elements; write $D_{2^{2}\cdot 3^{2}}$ for the corresponding divisor lattice.

Quadratic generation conjecture. The defining ideal $I_{{\mathcal R}_K(L)}$ is generated by quadrics if and only if at most one interval is $D_{2^{2}\cdot 3^{2}}$.

The preceding results show that sufficiently complicated interval configurations force relations of degree greater than two, while the conjecture proposes that having at most one interval of type $D_{2^{2}\cdot 3^{2}}$ is also sufficient for quadratic generation. The conjecture is presented as open in the source.