Truncated Vojta conjecture

Let $X$ be a smooth proper variety over a number field $k$, let $D$ be a normal crossings divisor, and let $A$ be an ample line bundle on $X$. Let $r$ be a positive integer and let $\epsilon>0$. Truncated Vojta conjecture. There is a proper Zariski-closed subset $Z\subset X$ containing $D$ such that

$$N_k^{(1)}(D,P)+d_k(k(P))\geq h_{K_X(D)}(P)-\epsilon h_A(P)-O(1).$$

The source states that Vojta showed this formulation is equivalent to the preceding conjecture, so it has the same status there.