The genuine mixed Schmidt conjecture for three weights

Let $\mathcal{D}$ be as above. For triples satisfying

$$0\leq i,j,k\leq 1,\qquad i+j+k=1,$$

let $\mathbf{Bad}_{\mathcal{D}}(i,j,k)$ be the set of $(x,y)\in\mathbb{R}^2$ for which there exists $c(x,y)>0$ such that

$$\max\left\{|q|_{\mathcal{D}}^{1/i},\|qx\|^{1/j},\|qy\|^{1/k}\right\}>c(x,y)q^{-1}\qquad\forall q\in\mathbb{N},$$

with the boundary cases interpreted as specified in the source. The genuine mixed Schmidt conjecture. For any $(i_1,j_1,k_1)$ and $(i_2,j_2,k_2)$ satisfying these conditions,

$$\mathbf{Bad}_{\mathcal{D}}(i_1,j_1,k_1)\cap\mathbf{Bad}_{\mathcal{D}}(i_2,j_2,k_2)\neq\emptyset.$$

The source explains that this interpolates between the classical Schmidt conjecture and the mixed Schmidt conjecture studied in the paper. It remains open in the stated generality.