Uniqueness of the optimal part number for dichromatic triangles

Prove that the set R_2(s) of even integers maximizing the dichromatic-triangle optimization function g_s(r) is a singleton for every integer s≥2 except s=27; equivalently, prove that the equation g_s(r)=g_s(r+2), with r=r_2(s), has no integer solutions for s≠27.

Background

The paper completely determines the dichromatic triangle problem for sufficiently large n once the optimizing set R_2(s) is known. It proves that R_2(s) contains at most two consecutive even integers and that the only observed tie is R_2(27)={2,4}. A remaining number-theoretic problem is to rule out all other ties, which would establish uniqueness of the candidate extremal Turán graph at the optimization level.

References

We strongly suspect that, apart from for $s=27$, the set contains a single element. This is a number theoretic statement: we conjecture that the equation $g_s(r)=g_s(r+2)$, or equivalently, $$ z{(r-1-a)(r+2)}(z+1){a(r+2)} = y{(r+1-b)r}(y+1){br} $$ where $z=\lfloor\frac{s}{r-1}\rfloor$, $y=\lfloor\frac{s}{r+1}\rfloor$, $a=s-(r-1)z$, $b=s-(r+1)y$, has no solutions for any integer $s \neq 27$, where $r=r_2(s)$.

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Section 'The dichromatic triangle problem' in Section 6, 'Concluding remarks'

We conjecture that $R_2(s)$ is a singleton unless $s=27$ but are unable to prove this.

— A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case  (2502.12291 - Gupta et al., 17 Feb 2025) in Section 1, subsection “Forbidding dichromatic triangles”; repeated in Section 6, subsection “The dichromatic triangle problem”