Trailing Betti numbers for Sally type semigroups S^e(m)

Fix the multiplicity $e$ and write $\beta_j(m):=\beta_j(S^e(m))$. Trailing Betti-number conjecture. For every $m\geq 2$, the Betti numbers satisfy

$$\beta_j(1)=\beta_j(m)\quad\text{for all }j\leq m-2,$$

and

$$\beta_j(m)=\beta_j(1)+\binom{e-m}{j+1-m}\quad\text{for }m-1\leq j\leq e-2.$$

This proposes that the Betti sequence of $S^e(m)$ agrees initially with that of the Gorenstein semigroup $S^e(1)$ and then differs by the displayed correction term; the status of the formula is not resolved in the supplied text.