The cross-characteristic field and transcendental conjecture for $On_p$

Let $On_p$ denote the ordinal system associated with the prime $p$. Call an element of $On_p$ an infinite field or a transcendental according to the paper's preceding definitions.

Cross-characteristic field and transcendental conjecture. If $\phi\in On_2$ is an infinite field, then $\phi$ is a field in $On_p$ for all primes $p$. If $\phi\in On_2$ is a transcendental, then $\phi$ is a transcendental in $On_p$ for all primes $p$.

This conjecture asserts that the field and transcendence pattern identified in $On_2$ persists across all prime characteristics, extending the pattern observed below $[\omega^{\omega^\omega}]$. The source gives no resolution of the conjecture.