  We study the cohomology of Lie superalgebra of vector fields on affine
super-spaces $\mathbb{A}^{m,n}$ with trivial coefficients. In this paper we
extend the methodology developed in the previous paper (arXiv:2210.16585) to
perform the calculation for arbitrary dimensions $(m, n)$ with $m \geq n$. This
combined with previously known results for other dimensions $(m, n)$ fully
settles the local problem of Gelfand-Fuchs cohomology with trivial coefficients
for super-manifolds. As part of the proof we also compute the cohomology of Lie
superalgebras $\mathfrak{gl}(m,n)$ with coefficients given by Schur functors of
the standard representation.
