Trailing Betti numbers for S^e(2,n)

Fix the multiplicity $e$ and write $\beta_j(a,b):=\beta_j(S^e(a,b))$. Trailing Betti-number conjecture for S^e(2,n). For $n\geq 6$,

$$\beta_j(2,n)=\beta_j(2,3)\quad\text{for all }j\leq n-5,$$

and

$$\beta_{n-4}(2,n)=\beta_{n-4}(2,3)+1.$$

Moreover, $S^e(2,e-1)$ is conjectured to be the closest to being Gorenstein in this family, with

$$\beta_j(2,e-1)=\begin{cases}\beta_j(2,3),&j\leq e-6,\beta_j(2,3)+1,&j=e-5,\beta_j(2,3)+3,&j=e-4,\beta_j(2,3)+2=3,&j=e-3.\end{cases}$$

The claim compares the Betti sequences with those of the Gorenstein case $S^e(2,3)$; no resolution status is supplied in the text.