Equal removable-hook counts do not replace the core test

The partition $(5)$. Its hook-length diagram is the single row $5,4,3,2,1$, so by [F1] the unique hook of length $3$ is $H_{1,3}(5)$, a horizontal rim hook of size $3$ spanning the nodes $(1,3),(1,4),(1,5)$ whose removal leaves the partition $(2)$; hence $(5)$ has exactly one removable $3$-hook. Its $\beta$-set is $X_{\lambda }=\{b_1\}=\{5\}$, a single bead on runner $2$ at row $1$; top-packing slides it to row $0$, so $X^{(3)}=\{2\}$ and $C_3((5))=P(\{2\})=(2)$, a $3$-core since $2<3$ and $(2)$ has hook lengths $2,1$.

The partition $(4,1)$. Its hook-length diagram is $5,3,2,1$ in the first row and $1$ in the second, so by [F1] the unique hook of length $3$ is $H_{1,2}(4,1)$, whose rim hook is the horizontal strip $(1,2),(1,3),(1,4)$ with complement $(1,1)$; hence $(4,1)$ also has exactly one removable $3$-hook. Its $\beta$-set is $X_{\mu }=\{b_1,b_2\}=\{5,1\}$, with $5$ on runner $2$ at row $1$ and $1$ on runner $1$ at row $0$; top-packing slides the bead $5$ to the position $2$, so $X^{(3)}=\{2,1\}$ and $C_3((4,1))=P(\{2,1\})=(1,1)$, a $3$-core since $(1,1)$ has hook lengths $2,1$.

The two partitions therefore have the same number (one) of removable $3$-hooks, but their $3$-cores are the distinct partitions $(2)$ and $(1,1)$ of size $2$; by [F3] the Specht modules $S_k^{(5)}$ and $S_k^{(4,1)}$ lie in different blocks of $k[S_5]$ for every splitting field $k$ of characteristic $3$. The top-packed abacus test of [F2] distinguishes the two configurations (one bead on runner $2$ versus beads on runners $2$ and $1$), while the local removable-hook count does not, so no such count can replace the core test.