The equality conjecture for $H(m,2)$ and $H_2(m,2)$

Let $H(m,2)$ and $H_2(m,2)$ denote the two quantities defined in the paper for integers $m\geq 3$. Equality conjecture.

$$H(m,2)=H_{2}(m,2)\quad\text{for all }m\geq 3.$$

Consequently, the paper's forbidden-configuration function satisfies

$$\operatorname{forb}(m,3,M)=2^m+m2^{m-1}+H(m,2),$$

and its asymptotic excess over $1$ is

$$\lim_{m\to\infty}\frac{\operatorname{forb}(m,3,M)}{m2^{m-1}}-1=\lim_{m\to\infty}\frac{H(m,2)}{m2^{m-1}}=\lim_{m\to\infty}\frac{H_{2}(m,2)}{m2^{m-1}}=\lambda(2).$$

The conjecture would identify the general quantity $H(m,2)$ with the recursively tractable quantity $H_2(m,2)$ and thereby determine the asymptotic constant for the $r=3$ forbidden-configuration problem. The preceding theorem establishes the corresponding exact recurrence and limit for $H_2(m,2)$, but the equality with $H(m,2)$ remains conjectural.