  Bernstein, Frenkel and Khovanov have constructed a categorification of tensor
products of the standard representation of $\mathfrak{sl}_2$ using singular
blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$. In earlier work, we
construct a positive characteristic analogue using blocks of representations of
$\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p > n$, with
zero Frobenius character, and singular Harish-Chandra character. In the present
paper, we extend these results and construct a categorical
$\mathfrak{sl}_k$-action, following Sussan's approach, by considering more
singular blocks of modular representations of $\mathfrak{sl}_n$. We consider
both zero and non-zero Frobenius central character. In the former setting, we
construct a graded lift of these categorifications which are equivalent to a
geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul
duality between two geometric categorificatons constructed in their work, and
resolve a conjecture of theirs. For non-zero Frobenius central characters, we
show that the geometric approach to categorical symmetric Howe duality by
Cautis and Kamnitzer can be used to construct a graded lift of our
categorification using singular blocks of modular representations of
$\mathfrak{sl}_n$.
