Griffiths' conjecture on Nevanlinna's second theorem

Let $X$ be a complex projective variety, let $A$ be an ample line bundle on $X$, and write

$$K_X:= \wedge^{\dim(X)}T_X^{\ast}$$

for its canonical line bundle. Let $D$ be a normal crossing divisor on $X$. For a holomorphic curve $f:\mathbb{C}\to X$, let $m_{f,D}(r)$ denote its proximity function and $T_{f,A}(r)$ and $T_{f,K_X}(r)$ its characteristic functions; write $\leq_{\operatorname{exc}}$ for an inequality holding outside an exceptional set of values of $r$.

Griffiths' conjecture. (1) For every holomorphic curve $f:\mathbb{C}\to X$ with Zariski-dense image,

$$m_{f,D}(r)+T_{f,K_X}(r)\leq_{\operatorname{exc}}O\bigl(\log^+T_{f,A}(r)\bigr)+o\bigl(\log r\bigr).$$

(2) For every $\epsilon>0$, there exists an algebraic subset $Z\nsubseteq X$ such that, for every holomorphic curve $f:\mathbb{C}\to X$ with $f(\mathbb{C})\not\subset Z$,

$$\forall C\in\mathbb{R},\quad m_{f,D}(r)+T_{f,K_X}(r)\leq_{\operatorname{exc}}\epsilon T_{f,A}(r)+C.$$

This is the conjectural form of Nevanlinna's second theorem for holomorphic curves. The supplied context does not state whether either formulation is known or remains open.