Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with
$d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with
$i+j=1$. We prove that the set of $x \in \RR$ for which there exists some
constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q
\qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt
games). Thus the intersection of any countable number of such sets is of full
dimension. In turn, this establishes the natural analogue of Schmidt's
conjecture within the framework of the de Mathan-Teuli\'e conjecture -- also
known as the `Mixed Littlewood Conjecture'.