Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?
$\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space
$R$ is compact as a module over $R \otimes R^{op}$
$R$ is not concentrated in degree 0
Motivation: I am looking for a smooth proper category with negative Hochschild cohomology and someone suggested $\mathit{Perf}(R)$ with $R$ satisfying the above if it existed. In fact they suggested to look for a finite CW complex $X$ with $H_*(\Omega X,\mathbb{F}_p)$ finite over $\mathbb{F}_p$ (then we can take $R=H_*(\Omega X,\mathbb{F}_p)$) but I have not found any of those either.
