Conjecture: The torus knot K5a2 has no friends

Establish whether the (2,5)-torus knot K5a2 has no non-isotopic knot K' such that the 0-surgery K5a2(0) is orientation-preservingly diffeomorphic to K'(0); equivalently, prove or disprove that K5a2 has no friends.

Background

In exploring minimal crossing-number friends, the authors note that some low-crossing knots—the unknot, trefoil (K3a1), figure-eight (K4a1), and K5a1—are known to have no friends. The next candidate that might have a friend is the 5-crossing torus knot K5a2 (which in their notation corresponds to the (2,5)-torus knot).

They explicitly state a conjecture that K5a2 has no friends, leaving open the existence of any non-isotopic knot sharing its 0-surgery. Resolving this would further clarify the landscape of low-complexity examples and constraints on 0-surgery equivalence among torus knots.