A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$ Is the following conjecture true?
Conjecture. If $r > s \ge 1$ are relatively prime integers such that
\begin{equation}
  (r-s)^4-1 \equiv 0\!\pmod{4r^2s},  \tag{1}
\end{equation}
then $r-s = 1$ or $2r > 3s$.
A brute-force computer search has so far found only solutions with $r-s=1$ and the two additional solutions $(r,s)=(10,3)$ and $(r,s)=(255,4)$. [n.b. Noam Elkies confirmed the conjecture up to $r = 3 \cdot 2^{22} > 1.25 \cdot 10^7$ using gp.]
I posted a partial proof on MSE, but got no help despite several upvotes and a bounty offer (which has since expired).
The motivation for the proof is the application of Vieta jumping to equations greater than the second degree, primarily as a method of attacking Thue equations. So although any proof of the conjecture would be nice, a completion of my partial proof would be preferred.
 A: You are attempting to generalize to cubics, a form of infinite descent sometimes used with the aid of quadratic polynomials. In short (see the link for a better description) one assumes a certain positive integer pair is the minimal solution of some problem, produces a quadratic polynomial they solve, adjusts it to get another, and that the new polynomial gives a smaller valid solution.
As I commented on MSE (before giving a possible approach which does not merit repeating): Your partial proof uses a cubic with a known integer root, $w_1,$ and then discusses the case that the other two roots are real (perhaps not integers). However for this problem they will never be real, so that entire line of argument does not seem  promising as a vehicle for generalizing the technique to polynomials of degree $3$ and higher. 
If that is a strong motivation, then perhaps another problem would be better.
The link also shows that around 20 years ago this technique solved a tricky Math Olympiad problem and, in following years, a number of the set olympiad problems have had nice solutions using it. I'm not sure how successful it has been with problems not designed to utilize it.  But perhaps it should be better known than it is.
