On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in $S_n$

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For a composition $lambda$ of $n$ our aim is to obtain reduced forms ‎for all the elements in the Kazhdan-Lusztig (right) cell containing ‎$w_{J(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $S_n$ corresponding to $lambda$‎. ‎We investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{J(lambda)}d$‎, ‎where $d$ is a prefix of ‎an element of minimum length in a $(W_{J(lambda)},B)$ double coset‎ ‎with the trivial intersection property‎, ‎$B$ being a parabolic subgroup ‎of $S_n$ whose type is `dualʹ to that of $W_{J(lambda)}$‎.