Some remarks about disjointly homogeneous symmetric spaces

Let 1 <= p < infinity. A symmetric space X on [0, 1] is said to be p-disjointly homogeneous (resp. restricted p-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from X (resp. characteristic functions) contains a subsequence equivalent in X to the unit vector basis of l(p). Answering a question posed in the paper (Hernandez et al. in Funct Approx 50(2): 215-232, 2014), we construct, for each 1 <= p < infinity, a restricted p-disjointly homogeneous symmetric space, which is not p-disjointly homogeneous. Moreover, we prove that the property of p-disjoint homogeneity is preserved under Banach isomorphisms.