Fuentes García's projective normality conjecture for polarized abelian varieties

Let $(X,L)$ be a general polarized abelian variety of type $(1,\dots,1,d)$ and dimension $g$.

Fuentes García's conjecture. $L$ is projectively normal if

$$d \geqslant 2^{g+1}-1.$$

Projective normality is subtle for polarizations of type $(1,\dots,1,d)$, since the first elementary divisor is $1$. The conjecture is stated as a proposed bound for general polarized abelian varieties; the supplied source does not establish its resolution.