Winning criterion for inhomogeneous badly approximable sets on lines

Let $(i,j)$ be non-negative real numbers with $i+j=1$. For $a,b\in\mathbb{R}$ with $a\ne0$, let ${\rm L}_{a,b}$ be the line $y=ax+b$. Let $\mathbf{Bad}(i,j)$ be the homogeneous weighted badly approximable set and $\mathbf{Bad}_{\bm\theta}(i,j)$ its inhomogeneous counterpart.

Winning criterion conjecture.

$$\mathbf{Bad}(i,j)\cap{\rm L}_{a,b}\ne\varnothing$$

if and only if, for every $\bm\theta\in\mathbb{R}^2$,

$$\mathbf{Bad}_{\bm\theta}(i,j)\cap{\rm L}_{a,b}\text{ is winning}.$$

The source calls this a plausible necessary-and-sufficient condition; it follows known optimal results in the rational-slope case but remains open in general.