Let $X$ and $Y$ be two topological spaces. We denote with $CX$ and $CY$ the sets of connected components and $QX$ and $QY$ the sets of quasicomponents for the corresponding spaces. It is well known that if $X$ and $Y$ are homotopic, then $|CX|=|CY|$ and $|QX|=|QY|$.
Is there a similar claim (or a known counterexample) when spaces $X$ and $Y$ are shape morphic?