Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also known as cochains with sphere spectrum coefficients).

It is known that (at least for simply connected spaces), the functor $X\mapsto S(X)$ from the $\infty$-category of spaces to $E_\infty$ rings is fully faithful, and its essential image can be nicely characterized by Joyal's "characteristic 1" generalization of Quillen's classical theorem about the functor $X\mapsto C^*(X, \mathbb{Q})$ from $\mathbb{Q}$-homotopy types to commutative DG rings over $\mathbb{Q}$. I seem to remember that this functor is nicely compatible with taking loops. More specifically, if $X$ is a pointed space then $S(X)$ has canonically an augmentation, and if $X$ is $3$-connected or more, we have $$S(\Omega(X)) = \operatorname{Tor}_{S(X)}(S, S)$$ ($S$ is a module via the augmentation). For a more general $E_\infty$ algebra spectrum $E$ with augmentation, write $\Omega(E) : = \operatorname{Tor}_{S(X)}(S, S).$ (The "Bar construction" in the sense of Lurie, http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf.) This will in general be a bialgebra.

More generally if $X$ is a topological monoid, then $S(X)$ is a co-monoid object in $E_\infty$ rings, i.e. an $(E_\infty, E_1)$ bi-algebra. If $X$ is an $E_\infty$-space then $S(X)$ is an $E_\infty,$ co-$E_\infty$ bialgebra (which is Hopf-like if $X$ admits homotopy inverses). Now $E_\infty$ spaces with homotopy inverses are (an equivalent $\infty$-category to) connective spectra. Furthermore, the two shift operations $X\mapsto \Omega(X)$ and $X\mapsto BX$ from $E_\infty$ spaces to $E_\infty$ correspond (again, if I am not mistaken) to the bar and cobar constructions on bialgebras, respectively, which I'll denote $\Omega( S(X))$ and $\Omega^{co}(S(X))$, respectively. In particular to any (bounded-below) spectrum $X$ one can associate an object $S(X)$ of (something like) the "category of connective bialgebras localized at maps of the form $H\mapsto \Omega^{co}(\Omega_{\ge 0} H)$".

My question is whether this functor (or a better functor of this sort) is compatible with Spanier-Whitehead duality.

In other words, say that $X$ is a connective spectrum of finite type (essentially, the loop spectrum of a finite CW complex), which is further simply connected and has homology in degrees $\le n$. Then the Spanier-Whitehead dual spectrum $X^\vee$ has homology in degree $\ge -n,$ and its $n+2$-fold suspension $$Y:=\Sigma^{n+2}X^\vee$$ is also a connective and simply-connected spectrum, which we can view as an $E_\infty$ space.

Question. Is there an algebraic relationship between the bialgebras $S(X)$ and $S(Y)$?

One can make the following naive guess.

Guess: the dual bialgebra $S(X)^\vee$ satisfies $$S(X)^\vee \cong \Omega^{n+2}S(Y).$$

This seems too optimistic (among other things, even infinite loop spaces corresponding to dualizable spectra might have infinite-dimensional cochains). But is there a way to make such a statement true? What about a statement of this sort for the functor of cochains with $\mathbb{Q}$-coefficients?