Conjectural counterexample to Murphy
p=3,      2849p = 8547 = 3*7*11*37
There is an elliptic curve
y2 + xy + y = x3 + x2 - 154x - 478
of conductor 8547, such that
fE = fF (mod 3) and
F has rank 2.
Maybe Sel3(E) = Sel3(F),
which would imply Sha(E) is "explained by a
jump in the rank of the Mordell-Weil"; equivalently,
Sha(E) is visible in J0(8547).
This is only a conjecture.