Normal generation conjecture for polarized abelian varieties of type $(1,\ldots,1,d)$

Let $(A,L)$ be a polarized abelian variety of dimension $g$. A polarization is of type $(d_1,\ldots,d_g)$ when its elementary divisors are $d_1\mid\cdots\mid d_g$. Normal generation conjecture. If $L$ is of type $(1,\ldots,1,d)$ with $d\geq 2^{g+1}-1$, then $L$ is normally generated. The statement concerns the complementary high-degree boundary case to the exceptional types discussed above and predicts projective normality for sufficiently large $d$.