Transcendence of the prime-indexed sequence modulo primes

Let $p_n$ denote the $n$th prime, and define

$$\pi(\boldsymbol{p})=\left(\pi(p)\bmod p\right)_p=\left(n\bmod p_n\right)_n\in\mathcal{A}.$$

Let $\mathcal{C}_{\mathcal{A}}$ denote the subring of algebraic elements of $\mathcal{A}$. Transcendence conjecture.

$$\pi(\boldsymbol{p})\in\mathcal{A}\setminus\mathcal{C}_{\mathcal{A}}.$$

The paper proves the weaker statement that $\pi(\boldsymbol{p})\in\mathcal{A}\setminus\mathcal{P}^0_{\mathcal{A}}$, so the conjecture asks for transcendence in the naive sense beyond the known exclusion from the indicated subring.