For any triple

of pair-wise coprime positive integers satisfying

,
we study the exponential Diophantine equation

. First, we conjecture that the equation has no positive solution other than

.
Second, we prove that our conjecture is true under
certain congruence conditions on
a,
b and
c.
The proof relies upon the theory of cubic residue
and several existing results concerning the generalized Fermat equation

.
Our result can be regarded as a relevant analogue to some results on Jesmanowicz'
conjecture concerning Pythagorean triples.