The diagonal recurrence conjecture for Betti numbers of squared-path cut complexes

Let $P_n$ be the path graph on $n$ vertices, let $P_n^2$ denote its square, and let $\beta_k(k+r)$ be the unique nonzero Betti number of the $(r-1)$-dimensional shellable cut complex $4\Delta_k(P_{k+r}^2)4$ for the squared path, where $r\ge 3$.

Diagonal recurrence conjecture. For each fixed $r\ge 3$ and $k\ge r+3$, these numbers satisfy

$$\beta(k,k+r)=\binom{r}{1}\beta(k-1,k-1+r)-\binom{r}{2}\beta(k-2,k-2+r)+\cdots+(-1)^{r-1}\binom{r}{r}\beta(k-r,k-r+r).$$

The known values are $\beta(3,n)=\binom{n-4}{2}$, $\beta(k,k+2)=0$, and $\beta(k,k+3)=\binom{k-1}{2}$ for $k\ge 3$. The first diagonal is already accounted for, while subsequent diagonals agree with identified integer sequences; the recurrence remains conjectural in general.