In my first question Lorenz attractor path-connected?, some are saying the Lorenz attractor $\mathscr L$ is not path-connected.

But suppose $x$ and $y$ are two points in different path components of the attractor. By Zorn's Lemma there is a subcontinuum of $\mathscr L$ which is irreducible between $x$ and $y$ (no proper subcontinuum of it contains these two points). I am having hard time imagining any irreducible subcontinua in $\mathscr L$ other than paths...

Are the "butterfly's wings" connected, individually?

EDIT: It seems a lot of people are not understanding my question. I want you to demonstrate an irreducible subcontinuum of $\mathscr L$ which is not an arc. If it is not path-connected, then there must be such an example. And the example cannot be $\mathscr L$ itself, because $\mathscr L$ is the union of two proper subcontinua, and these sub continua extend to points in their complements by small connecting arcs.