Eventual vanishing conjecture for $M(a,b)$ in SALIQUANT

For positive integers $a$ and $b$, define $M(a,b)=m$ by

$$\mathcal{SG}((2a+1)2^b)=f(a,b,m).$$

Eventual vanishing conjecture for $M(a,b)$. For each fixed $a>0$, there is a sufficiently large $b$ such that

$$M(a,b)=0,$$

and consequently

$$\mathcal{SG}((2a+1)2^b)=(2a+1)2^{b-1}-a-1.$$

The conjecture is motivated by the sparsity of the nonzero entries observed in each column of the experimental table for $M(a,b)$. It predicts that every fixed column eventually consists entirely of zeros, although the available data are limited because the relevant values grow exponentially with $b$.