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 Teulié 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.
