Mean-value conjecture for character sums in Pascal's triangle modulo p

Let $p$ be prime, let $\chi$ range over characters modulo $p$, and let $\chi_0$ be the principal character. Define

$$A_p=\{\phi_\chi(p):\chi(-1)=1,\ \chi\neq\chi_0\},\qquad B_p=\{\phi_\chi(p):\chi(-1)=-1\}.$$

Let $\mu_{A_p}$ and $\mu_{B_p}$ be the means of these two sets.

Mean-value conjecture. As $p$ tends to infinity,

$$\mu_{A_p}\sim 3p,\qquad \mu_{B_p}\sim 2p.$$

The conjecture is motivated by probabilistic calculations concerning the fundamental domain of Pascal's triangle modulo $p$. The source notes that dependence among the values $\phi_\chi(p)$ prevents a direct application of the central limit theorem, and gives no resolution of the conjecture.