It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space.

Let $X$, $Y$ be two injective Banach spaces such that,

• There exists an into linear isometry $i:X\to Y$, and
• There exists an into linear isometry $j:Y\to X$.

Q: Does there exist a surjective isometry $X\to Y$?

Dually, if $M_X$, $M_Y$ are compact Hausdorff extremally disconnected spaces such that there exist surjective continuous maps $j^{*}:M_X\to M_Y$, $i^{*}:M_Y\to M_X$, then

$\textbf{Q}^*\textbf{:}$ Are $M_X$ and $M_Y$ homeomorphic?