  We prove a metrical result on a family of conjectures related to the
Littlewood conjecture, namely the original Littlewood conjecture, the mixed
Littlewood conjecture of de Mathan and Teuli\'e and a hybrid between a
conjecture of Cassels and the Littlewood conjecture. It is shown that the set
of numbers satisfying a strong version of all of these conjectures is large in
the sense of Hausdorff dimension restricted to the set of badly approximable
numbers.
