Cardinal p and a theorem of Pelczynski

We show that it is consistent that for some uncountable cardinal k, all compactifications of the countable discrete space with remainders homeomorphic to $D^k$ are homeomorphic to each other. On the other hand, there are $2^c$ pairwise non-homeomorphic compactifications of the countable discrete space with remainders homeomorphic to $D^c$ (where c is the cardinality of the continuum).