Schmidt's conjecture on intersecting weighted badly approximable sets

Let an $n$-dimensional weight be a vector $\boldsymbol{\mathbf{r}}=(r_1,\dots,r_n)$ with $r_i\geq 0$ and $r_1+\cdots+r_n=1$. A vector $\mathbf{x}\in\mathbb{R}^n$ is $\boldsymbol{\mathbf{r}}$-weighted badly approximable if there exists $c>0$ such that, for every $(p_1,\dots,p_n,q)\in\mathbb{Z}^{n+1}$ with $q\neq 0$, $\max_{1\leq i\leq n}|q|^{r_i}|qx_i+p_i|\geq c$; denote the set of such vectors by $\mathrm{\mathbf{Bad}}(\boldsymbol{\mathbf{r}})$. Schmidt's conjecture. The two sets of weighted badly approximable vectors satisfy

$$\mathrm{\mathbf{Bad}}(1/3,2/3)\cap\mathrm{\mathbf{Bad}}(2/3,1/3)\neq\emptyset.$$

This conjecture concerns the intersection of badly approximable sets for different weights; although each weighted badly approximable set has Lebesgue measure zero and full Hausdorff dimension, the nonemptiness of this particular intersection was posed by Wolfgang M. Schmidt in 1982 and remains unresolved in the supplied source.