  We deal with $g$-dimensional abelian varieties $X$ over finite fields. We
prove that there is an universal constant (positive integer) $N=N(g)$ that
depends only on $g$ that enjoys the following properties. If a certain
self-product of $X$ carries an exotic Tate class then the self-product
$X^{2N}$of $X$ also carries an exotic Tate class. This gives a positive answer
to a question of Kiran Kedlaya.
