Stability of initial Betti numbers for S^e(m,m+1)

Fix the multiplicity $e$. Betti-number stability conjecture. For every $m\geq 4$, the Betti numbers of $S^e(m,m+1)$ and $S^e(m+1,m+2)$ agree through the indicated range; equivalently, for every $n\geq m$,

$$\beta_j(n,n+1)=\beta_j(m,m+1)\quad\text{for all }j\leq m-3.$$

Thus the first $m-3$ Betti numbers are predicted to be independent of $n$ along this family; the supplied text gives no resolution status.