Non-degeneracy conjecture for partial heights

Let $K=\mathbb{C}(B)$ be the function field of a smooth projective curve $B$ over $\mathbb{C}$. Let $X$ be a projective variety over $K$ that does not contain any possibly singular rational curve. Let $\pi:{\mathcal {X}}\to B$ be an integral model of $X$ over $B$. Let $h_L:X(K)\to\mathbb{R}$ be a Weil height function associated to an ample line bundle $L$ on $X$, and let $h_{(D,\omega)}:X(K)\to\mathbb{R}$ be a partial height function associated to a strictly positive pair $(D,\omega)$ on ${\mathcal {X}}$, where $D$ is an open disc in $B$. For a sequence $\{x_n\}_{n\geq 1}$ in $X(K)$, non-degeneracy conjecture. if $h_L(x_n)$ converges to infinity, then $h_{(D,\omega)}(x_n)$ converges to infinity.