  For a ribbon knot, it is a folk conjecture that the rank of its knot Floer
homology must be 1 modulo 8, and another folk conjecture says the same about
reduced Khovanov homology. We give the first counter-examples to both of these
folk conjectures, but at the same time present compelling evidence for new
conjectures that either of these homologies must have rank congruent to 1
modulo 4 for any ribbon knot. We prove that each revised conjecture is
equivalent to showing that taking the rank of the homology modulo 4 gives a
homomorphism of the knot concordance group. We check the revised conjectures
for 2.4 million ribbon knots, and also prove they hold for ribbon knots with
fusion number 1.
