Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?

1. $\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space

2. $R$ is compact as a module over $R \otimes R^{op}$

3. $R$ 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.

Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?

1. $\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space

2. $R$ is compact as a module over $R \otimes R^{op}$

3. $R$ 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.

Edit: As Shaul Barkan explained to me, such an example cannot come from a space X as described above, because the conditions will imply that $H_*(\Omega X,\mathbb{F}_p)$ is in degree 0.

Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?

1. $\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space

2. $R$ is compact as a module over $R \otimes R^{op}$

3. $R$ is not concentrated in degree 0

Motivation: I am looking for a smooth proper category with negative Hochschild cohomology and someone suggested $Perf(R)$ 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.

Are there any connective $E_1$ rings $R$ over $\mathbb{F}_p$ satisfying the following?

1. $\pi_*(R)$ is a finite dimensional $\mathbb{F}_p$ vector space

2. $R$ is compact as a module over $R \otimes R^{op}$

3. $R$ 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.