The super Alternative Daugavet property, unconditional bases and SCD geometry

We answer negatively Question 6.4 posed by Langemets, L\~oo, Mart\'in, Perreau and Rueda Zoca concerning the existence of infinite-dimensional Banach spaces with a 1-unconditional basis satisfying the super Alternative Daugavet property. We also address two recent questions posed by L\~oo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable weak $\pi$-base, yielding a partial positive answer to Question 5.1. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.