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In the seminal monograph "Theory of retracts", Borsuk raised the followingquestion: suppose two compact ANR's are $h$--equal, i.e. mutually homotopydominate each other, are they homotopy equivalent? The current paper approachesthis question in two ways. On one end, we provide conditions on the fundamentalgroup which guarantee a positive answer to the Borsuk question. On the otherend, we construct various examples of compact $h$--equal, not homotopyequivalent continua, with distinct properties. The first class of theseexamples has trivial all known algebraic invariants (such as homology, homotopygroups etc.) The second class is given by $n$--connected continua, for any $n$,which are infinite $CW$--complexes, and hence ANR's, on a complement of apoint.

On the Borsuk conjecture concerning homotopy domination