Nakayama's numerical-dimension growth conjecture

Let $X$ be a smooth projective variety and let $D$ be a pseudo-effective $\mathbb R$-divisor on $X$. Nakayama's numerical-dimension growth conjecture. There exist a positive integer $m_0$, a positive rational number $C$, and an ample Cartier divisor $A$ on $X$ such that

$$Cm^{\kappa_\sigma(X,D)}\leq \dim H^0\left(X,\mathcal O_X\left(\lfloor mm_0D\rfloor+A\right)\right)$$

holds for every sufficiently large positive integer $m$. The paper says that proving this would imply Nakayama's original inequality on $\kappa_\sigma$; no resolution is stated.