The third approximation claim for ν₂(S(m,5))

Let $\operatorname{Err}_2(m,5)$, $x_2(m)$, $\beta(m)$, and $f_3(m)$ be defined by the preceding approximation scheme. In particular,

$$\beta(m)=\alpha_m+(-1)^{m+1}\lambda_3(m),$$

and

$$f_3(m)=4^{1-\lambda_3(m)}\left\lfloor\frac{m+2}{3}\right\rfloor+\lambda_3(m)\left(85\lambda_3(m)+8\lambda_2(m+1)+2\lambda_3(m+1)\right).$$

The third approximation claim. For most $m\in\mathbb{N}$,

$$\operatorname{Err}_2(x_2(m),5)=(-1)^{\beta(x_2(m))}\nu_2(f_3(x_2(m))).$$

The source presents this as an empirical “final improvement,” not as a formally stated conjecture; its resolution is not given.