Finite-difference conjecture for Ext dimensions of complete intersections

Let $R$ be a complete intersection ring with $k$ relations and $n$ variables, where $k\geq 1$. For any map $f:\mathbb{N}\longrightarrow\mathbb{N}$, define $\Delta^0f(m)=f(m)$ and $\Delta^{r+1}f(m)=\Delta^rf(m+1)-\Delta^rf(m)$. Set $f(m)=\operatorname{Ext}_R(\mathbb{C},\mathbb{C})(m)$, with $\operatorname{Ext}_R(\mathbb{C},\mathbb{C})(m)$ denoting the degree-$m$ Ext group. Finite-difference conjecture. The following hold:

• If $k=1$, then for every $m\geq n-k$,

$$\Delta^{k-1}\bigl(\operatorname{dim}\operatorname{Ext}_R(\mathbb{C},\mathbb{C})(m)\bigr)=2^{n-k}.$$

• If $k\geq 2$, then for every $m\geq n-(k+1)$,

$$\Delta^{k-1}\bigl(\operatorname{dim}\operatorname{Ext}_R(\mathbb{C},\mathbb{C})(m)\bigr)=2^{n-k}.$$

The preceding cases $k=1,2,3$ motivate this numerical pattern for larger $k$, but no resolution is given in the supplied text.