Fix $p=2$ and let $D(n)$ be a spectrum which filters the Eilenberg-Moore spectrum $H\mathbb{Z}/2$, i.e. $\mathrm{colim}\ D(n)=H\mathbb{Z}/2$. This spectrum can be considered as the cofibre of $Sp^{2^{n-1}}S^0\to Sp^{2^n}S^0$ with $Sp^kS^0$ being the $k$-th symmetric power of the sphere spectrum.

We know $D(n)$ maps to a Thom spectrum, $H\mathbb{Z}/2$, and I think we can identify it with stable summand of a Thom spectrum. I wonder whether or not if it is known to be a Thom spectrum itself?!

I will be happy if, at the prime $2$, we can find a map $X\to \Omega^2S^3$ so that the Thom spectrum of the composition $X\to\Omega^2S^3\to BO$ is $D(n)$ or some (de-)suspension of it. Here, the map $\Omega^2S^3\to BO$ is the extension of $S^1\to BO$, representing a generator of $\pi_1BO$, to a double loop map, using the infinite loop structure on $BO$ provided by the Bott periodicity; it is known due to Mahowald that the Thom spectrum of $\Omega^2S^3\to BO$ is $H\mathbb{Z}/2$.