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Bayer, Denker, Jeli\\'c Milutinovi\\'c, Sundaram, and Xue proved that the $k$-cut complex of the squared path $P_n^2$ is shellable for $n\\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along every diagonal $n-k=r$. We prove the recurrence by giving the exact formula $\\beta(k,n)=\\binom{n-1}{k-1}-\\sum_{j=0}^{\\min\\{k-1,n-k\\}}\\binom{k-1}{j}(n-k-j+1)+(n-k)$ for $r=n-k\\ge3$. 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