Algebraic independence of distinct-prime valuation generating functions at independent points

Let $T_p(z)$ denote the valuation generating function associated with a prime $p$. Let $p_1,p_2,\dots,p_m$ be distinct primes, and let $\alpha_1,\alpha_2,\dots,\alpha_m$ be non-zero algebraic numbers with $|\alpha_i|<1$ for all $i$, and assume that $\alpha_1,\alpha_2,\dots,\alpha_m$ are multiplicatively independent. Independent-point algebraic independence conjecture. The values

$$T_{p_1}(\alpha_1),\, T_{p_2}(\alpha_2),\, \dots,\, T_{p_m}(\alpha_m)$$

are algebraically independent over $\overline{\mathbb{Q}}$. This conjecture concerns simultaneous algebraic independence across distinct primes and multiplicatively independent algebraic evaluation points; the source states that its proof remains beyond current techniques.