Circular flow number and oriented cycle double cover

Conjecture    Suppose G does not contain the Petersen graph as minor. Phi_c(G) <=  p/q if and only if  G has an oriented cycle double cover, consisting of cycles C(0), C(1), ..., C(p-1), such that if the underline graphs of C(i) and C(j) share a common edge then q <= |i-j| <=p-q.

The `if' part is easy to prove (the Petersen minor free condition is not needed). The `only if' part is true for p/q =4, 3, 2, (2k+1)/k (again the Petersen minor free condition is not needed).  If the oriented version 5-cycle double cover conjecture is true, then the above conjecture is true for p/q = 5 as well (again, the Petersen minor free condition is not needed).  However, there are example graphs that show that the above conjecture would not be true (in preparation) for some graphs which contains the Petersen graph as a minor.

It would be interesting to prove this conjecture for any other rational p/q.