  We propose a method of constructing abelian envelopes of symmetric rigid
monoidal Karoubian categories over an algebraically closed field $\bf k$. If
${\rm char}({\bf k})=p>0$, we use this method to construct generalizations
${\rm Ver}_{p^n}$, ${\rm Ver}_{p^n}^+$ of the incompressible abelian symmetric
tensor categories defined in arXiv:1807.05549 for $p=2$ and by Gelfand-Kazhdan
and Georgiev-Mathieu for $n=1$. Namely, ${\rm Ver}_{p^n}$ is the abelian
envelope of the quotient of the category of tilting modules for $SL_2(\bf k)$
by the $n$-th Steinberg module, and ${\rm Ver}_{p^n}^+$ is its subcategory
generated by $PGL_2(\bf k)$-modules. We show that ${\rm Ver}_{p^n}$ are
reductions to characteristic $p$ of Verlinde braided tensor categories in
characteristic zero, which explains the notation. We study the structure of
these categories in detail, and in particular show that they categorify the
real cyclotomic rings $\mathbb{Z}[2\cos(2\pi/p^n)]$, and that ${\rm Ver}_{p^n}$
embeds into ${\rm Ver}_{p^{n+1}}$. We conjecture that every symmetric tensor
category of moderate growth over $\bf k$ admits a fiber functor to the union
${\rm Ver}_{p^\infty}$ of the nested sequence ${\rm Ver}_{p}\subset {\rm
Ver}_{p^2}\subset\cdots$. This would provide an analog of Deligne's theorem in
characteristic zero and a generalization of the result of arXiv:1503.01492,
which shows that this conjecture holds for fusion categories, and then moreover
the fiber functor lands in ${\rm Ver}_p$.
