Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$ where $A$ is the Steenrod algebra and $Q_n$ is the Adams Margolis element which generates the exterior algebra $E(Q_n)$. In particular, $$ Q_2 = Sq^7 + Sq^6Sq^1 + Sq^5 Sq^2 + Sq^4 Sq^2 Sq^1.$$

If there is a spectrum X such that as an $A$-module $$ H^*(X) = A//E(Q_2),$$ then can we conclude that $X$ is weak equivalent to $k(2)$? In other words, is the homotopy type of $k(2)$ uniquely determined by its $A$-module structure?

[I am aware of the results in the paper, where it is proved that as an $A_{\infty}$ ring spectrum $K(n)$ as well as $k(n)$ is essentially unique (correct me if I am wrong).]

Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$ where $A$ is the Steenrod algebra and $Q_n$ is the Adams Margolis element which generates the exterior algebra $E(Q_n)$. In particular, $$ Q_2 = Sq^7 + Sq^6Sq^1 + Sq^5 Sq^2 + Sq^4 Sq^2 Sq^1.$$

If there is a spectrum X such that as an $A$-module $$ H^*(X) = A//E(Q_2),$$ then can we conclude that $X$ is weak equivalent to $k(2)$? In other words, is the homotopy type of $k(2)$ uniquely determined by its $A$-module structure?

[Note that my question is different from the results in the paper, where it is proved that as an $A_{\infty}$ ring spectrum $K(n)$ as well as $k(n)$ is essentially unique (correct me if I am wrong).]