Polynomial formulas 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 $\beta(k,n)$ be the unique nonzero Betti number of the cut complex $\Delta_k(P_n^2)$.

Polynomial formula conjecture. For $k=4$ and $k=5$, the Betti numbers are conjectured to satisfy

$$\beta(4,n)=3+8\binom{n-7}{1}+6\binom{n-7}{2}+\binom{n-7}{3},\qquad n\ge 7,$$ $$\beta(5,n)=6+20\binom{n-8}{1}+21\binom{n-8}{2}+7\binom{n-8}{3}+\binom{n-8}{4},\qquad n\ge 8.$$

These formulas extend the computed Betti-number data for the squared path; no proof or resolution is supplied here.