On $k$-accessibility of the core of $TU$-cooperative game

In the paper, a strengthening of the core-accessibility theorem by the author is proposed. The results obtained demonstrate that for any $k \geq 1,$ and for any imputation $x$ outside of the nonempty core, a $k$-monotonic sequential improvement trajectory $\{x_r\}_{r=0}^{\infty}$ with $x_0 = x$ exists, which converges to some element of the core. Here, $k$-monotonicity means that for any $r > 0,$ an imputation $x_r$ dominates any preceding imputation $x_{r-m}$ with $r \geq m$ and $m \in [1, k].$ Note that the core-accessibility theorem, mentioned above, was established for the case $k = 1$.
To show that $TU$-property is essential to provide $k$-accessibility of the core, we propose an example of $NTU$-cooperative game $G$ with a "black hole" being a closed subset $B \subseteq G(N)$ that doesn't intersect the core $C(\alpha_G)$ and contains all the sequential improvement trajectories originating at any point $x \in B$.