Are there perverse sheaves on abelian varieties with small Euler characteristic?

Question

Let $A$ be a simple abelian variety of dimension $g$. Let $K$ be an irreducible perverse sheaf on $A$. We know that $\chi(A,K)\geq 0$. (Corollary 1.4 of Franecki and Kapranov.) How small can $\chi(A,K)$ be?

There are some trivial examples. If $K$ is a lisse sheaf on all of $A$ placed in degree $g$, then $\chi(A,K)=0$. If $K$ is a skyscraper sheaf, then $\chi(A,K)=1$.

Beyond these trivial examples, if $A$ is the Jacobian of a curve $C$ of genus $g$ and $K$ is the constant sheaf on $C$, placed in degree $-1$ and pushed forward to $A$, then $\chi(A,K)=2g-2$.

I want to know if there is any nontrivial example with smaller Euler characteristic:

Does there exist a simple abelian variety $A$ of dimension $g$ and an irreducible perverse sheaf $K$ on $A$ such that $K$ is not lisse, $K$ is not a skyscraper sheaf, and $\chi(A,K)<2g-2$?

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We know every irreducible perverse sheaf of Euler characteristic $0$ or $1$ is trivial in this sense. (Proposition 10.1 of Kramer and Weissauer.) So it is equivalent to ask:

Does there exist a simple abelian variety $A$ of dimension $g$ and an irreducible perverse sheaf $K$ on $A$ such that $2 \leq \chi(A,K) <2g-2?$

One way to construct a perverse sheaf on an abelian variety is to take a smooth variety $X$ and a finite morphism $f: X\to A$ and pushforward the constant sheaf in degree $-\dim(X)$ along $f$. So a subquestion is:

Does there exist a simple abelian variety $A$ of dimension $g$, a smooth non-abelian variety $X$ of dimension $d$, and a finite map $f:X \to A$, such that $(-1)^{d} \chi(X) < 2g-2$?

Answer 1

Mihnea Popa explained this lower bound to me:

If $f: X \to A$ is a finite morphism (or weaker conditions are possible), then we have generic vanishing for every Hodge number of $X$ but $h^{p,q}$ for $p+q=d$. So $(-1)^d \chi(X)$ is just the sum of the generic values of $h^{p,q}$ for $p+q=d$. In particular, it is at least twice the generic value of $h^{0,d}$, which is equal to the arithmetic Euler characteristic of $\omega_X$. A lower bound for this is the codimension of $X$ as long as $A$ is simple or $X$ has no irrational pencils (slight variant on Theorem A of Pareschi and Popa). This gives a lower bound of $2g-2d$.

As far as I know this should hold for any perverse sheaf that arises from any geometric construction, because we should always be able to put a Hodge structure on it (making it a mixed Hodge module) and use the same argument.

I would be interested in further lower bounding arguments or upper bounding examples. The largest gap between the lower bound and the examples I know how to construct occurs for hypersurfaces, where the lower bound is $2$ and the best example I know about it is the theta divisor of a Jacobian, which has Euler characteristic $ \left( \begin{array}{c} 2g-2 \\ g-1\end{array}\right)$, or about $4^g$.