Existence conjecture for D-optimal rhombic designs

Let $K$ be the dimension, let $D$ be the covariance matrix of the random coefficients, and let $(d_0,d_1,d_2)^T\in\mathcal{C}_K$ denote its parameter vector. A rhombic design is an invariant design from the class studied for the multiple linear regression model on the hypercube, and a design is $D$-optimal if it maximizes the determinant of its information matrix under covariance matrix $D$. Existence conjecture. For even $K$, there is a $D$-optimal rhombic design for all

$$(d_0,d_1,d_2)^T\in\mathcal{C}_K.$$

For odd $K$, there is a $D$-optimal rhombic design for all

$$(d_0,d_1,d_2)^T\in\mathcal{C}_K\quad\text{with}\quad d_2\leq\frac{d_1}{2}.$$

The conjecture records the observed existence pattern for $D$-optimal rhombic designs: existence was found for all parameter values in even dimensions, whereas in odd dimensions it was established only in the stated region. Whether such designs exist for all remaining parameter values when $K$ is odd is left open.