  In 1998 Kleinbock conjectured that any set of weighted badly approximable
$d\times n$ real matrices is a winning subset in the sense of Schmidt's game.
In this paper we prove this conjecture in full for vectors in $\mathbf{R}^d$ in
arbitrary dimensions by showing that the corresponding set of weighted badly
approximable vectors is hyperplane absolute winning. The proof uses the Cantor
potential game played on the support of Ahlfors regular absolutely decaying
measures and the quantitative non-divergence estimate for a class of fractal
measures due to Kleinbock, Lindenstrauss and Weiss. To establish the existence
of a relevant winning strategy in the Cantor potential game we introduce a new
approach using two independent diagonal actions on the space of lattices.
