Schmidt's intersection conjecture for badly approximable sets

For weights $(i,j)$ with $i,j\ge 0$ and $i+j=1$, let $\mathbf{Bad}(i,j)$ denote the corresponding weighted badly approximable set in $\mathbb{R}^2$.

Schmidt's conjecture.

$$\mathbf{Bad}\left(\tfrac13,\tfrac23\right)\cap\mathbf{Bad}\left(\tfrac23,\tfrac13\right)\neq\varnothing.$$

This conjecture of Schmidt was resolved by a stronger theorem concerning intersections with vertical lines. It is therefore solved.