Algebraic independence conjecture for finite Bernoulli numbers and Fermat quotients

For $k\ge0$, let

$$\gb_k=\left(\frac{B_{p-k}}{k}\pmod p\right)_{p>k}$$

be the $\calA_1$-Bernoulli numbers, and for $k\ge2$ let

$$q_k=\left(\frac{k^{p-1}-1}{p}\pmod p\right)_{p>k}$$

be the $k$th $\calA_1$-Fermat quotient. Set $\gb_1=1$. Suppose $n_1,\dots,n_r\in\N$ and $\log n_1,\dots,\log n_r$ are $\Q$-linearly independent. Algebraic independence conjecture. The elements

$$\bigcup_{k=0}^{\infty}\{\gb_{2k+1}q_{n_1},\dots,\gb_{2k+1}q_{n_r}\}$$

are algebraically independent over $\Q$ in $\calA_1$. This conjecture is a finite analogue of the corresponding algebraic-independence expectation for products of odd zeta values and logarithms; it is not resolved in the source.