Xie--Yuan's non-degeneracy conjecture for partial heights

Let $K=\mathbb C(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety over $K$ containing no rational curve. Let $h_L$ be a Weil height attached to an ample line bundle on $X$, and let $h_{(D,\omega)}$ be a partial height attached to a strictly positive pair $(D,\omega)$, where $D$ is an open disc in $B$. Xie--Yuan's non-degeneracy conjecture. If a sequence ${x_n}\subset X(K)$ satisfies

$$h_L(x_n)\longrightarrow\infty,$$

then

$$h_{(D,\omega)}(x_n)\longrightarrow\infty.$$

The conjecture asserts that positive partial heights detect every sequence of unbounded full height on varieties without rational curves; it is cited as the source of the proof strategy and remains open as stated.