On the connected sums of the $(2,1)$-cable of the figure eight knot

We show that the 3-fold (resp. 6-fold) connected sum of the $(2,1)$-cable of the figure-eight knot cannot bound a smooth null-homologous disk in a punctured $S^2\times S^2$ (resp. in a punctured #_2 S^2 \times S^2. This result is obtained using a real version of the $10\mathrm{/}8$-inequality established by Konno, Miyazawa, and Taniguchi.