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Let $//$ denote the Hopf algebra quotient. We know that:

$$HF_{2}^*(ko) \simeq A//A(1)$$

$$HF_2^*(tmf) \simeq A//A(2)$$

By Hopf invariant one, we know there is no $X$ such that $HF_2^*(X) \simeq A//A(3)$.

I ask for the next best thing. What are some examples of $X$ for $n \geq 3$ such that

$$HF_2^*(X) \otimes A(n) = \bigoplus_i A \text{, or more generally, } HF_2^*(X) \text{ contains } A//A(n) \text{?} $$

I have heard that Stong has some results along the lines of $X = MO\langle 9 \rangle$ contains $A//A(3)$, and in general $MO\langle 6+n \rangle$ contains $A//A(n)$ for $n \geq 2$. However, I was not able to find the source.

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    $\begingroup$ Bahri and Mahowald showed that the Thom class gives a monomorphism $A//A(n) \to MO\langle N\rangle$ (where $N$ is the vector fields on spheres number for $n+1$). $\endgroup$
    – Dylan Wilson
    Commented Nov 17, 2017 at 10:30

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