Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map

$1+3: \Omega S^{2} \to BG_{3}$

where $1 +3$ is a unit in $3$-adics. Here $G_{3}$ is the unit component of $\Omega^{\infty}S_{3}$, where $S_{3}$ is the $3$-adic sphere spectrum.

If we look at the $H^{*}(T; \mathbb{Z}/3)$ as a mudule over Steenrod algebra, does the 4th cell connect to the bottom by $\mathscr{P}^{1}$?

I also want to extend my question for other primes except $2$. Prime $2$ I managed to do, from the knowledge of characteristic classes at $2$, but other primes seems impossible!