Normal generation conjecture for polarized abelian varieties of type $(1,3,\ldots,3,6)$

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 $(1,3,\ldots,3,6)$, then $L$ is not normally generated. This extends the open cases identified in dimensions $3$ and $4$, namely types $(1,3,6)$ and $(1,3,3,6)$; the conjecture predicts that these polarizations do not give projectively normal embeddings.