Chris Godsil informed me that two graphs discussed on p.291 and shown on p.292 of [1] disprove the conjecture stated in the May 2003 problem. These examples have been found by Gordon Royle. One of them is one of the Blanusa snarks on 18 vertices. The other one is a hypohamiltonian graph on 22 vertices.
References:
[1] C. Godsil, G. Royle, Algebraic Graph Theory, Springer, 2001.