Conditional plasticity of the unit ball of the ${\mathrm{\ell }}_{\mathrm{\infty }}$-sum of finitely many strictly convex Banach spaces

We prove that for any ${\mathrm{\ell }}_{\mathrm{\infty }}$-sum $Z={⨁}_{i\in [n]}{X}_i$ of finitely many strictly convex Banach spaces $(X_i{)}_{i\in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\text{\hspace{0.17em}⁣}:B_Z\to B_Z$ is an isometry, by constraining the componentwise behavior of the inverse $g=f^{-1}$ with a theorem admitting a graph-theoretic interpretation. We also show that if $X,Y$ are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X\to B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type $B_{X^{\mathrm{\prime }}}\to B_{X^{\mathrm{\prime }}}$ for some Banach space $X^{\mathrm{\prime }}$, and that a homeomorphic 1-Lipschitz non-isometry of type $B_X\to B_X$ restricts to a homeomorphic 1-Lipschitz non-isometry of type $B_S\to B_S$ for some separable subspace $S\le X$.