1. So, far as I understand it (which I don't), Lunnon's latest (3rd) proof's key new idea, is that the points at knight-distance <=d to the origin, form a convex set, i.e. are precisely those within a certain convex region? His lemma 6? 2. (Incidentally, roughly that idea was in my head back when I devised my conjectural distance formula in first place.) 3. Using this lemma, one does not need to minimize over 8 choices, but rather over at most 4. 4. Unfortunately, the claim (1) above is false. So for that reason, plus several things in Lunnon's latest proof which literally make no sense (e.g. the statement of lemma 6 actually makes no sense as written in combination with defn 4) I'm sorry to report that I must again denounce this as a non-proof. On the bright side, however, the idea of (3) would be useful. Even if so, though, I think it's still going to be messy since although 4<8, we still have 4>1. Also, about Lunnon's calculation of "maximum number of shortest paths" finding "growth like d^2 * 2^d" I suspect an exact formula for this can be found using binomial coefficients and a finite number of cases.