Conjecture on arithmetic hyperbolicity for divisors in a nef cone

Let $X$ be a projective variety defined over a number field $k$, of dimension $n$. Let $E_1,\ldots,E_r$ be nef Cartier divisors on $X$ such that $\sum_{j=1}^rE_j$ is ample. Let $D_1,\ldots,D_q$ be non-zero effective, possibly reducible, Cartier divisors in general position, and let

$$D=\sum_{i=1}^qD_i.$$

Suppose that

$$D_i\equiv\sum_{j=1}^r a_{i,j}E_j$$

for non-negative real numbers $a_{i,j}$, and write $P_i=(a_{i,1},\ldots,a_{i,r})\in\mathbb{R}^r$. Assume that for every proper subset $T$ of the standard basis vectors $\{e_1,\ldots,e_r\}\subset\mathbb{R}^r$, at most $|T|\left\lfloor q/r\right\rfloor$ of the vectors $P_1,\ldots,P_q$ are supported on $T$. The nef-cone conjecture. If $q\geq r(n+1)+1$, then $X\setminus D$ is arithmetically quasi-hyperbolic; if $q\geq 2nr+1$, then $X\setminus D$ is arithmetically hyperbolic.