Finite-exception conjecture for surjectivity of the maps $\tau^{\mathfrak p}$

Let $K$ be a totally real number field of degree $n$. For an odd prime $p$ unramified in $K/\mathbb{Q}$ and a place $\mathfrak{p}$ above $p$, let $\tau^{\mathfrak p}$ be the map appearing in the source's criterion. Finite-exception conjecture. There exist at most finitely many primes $p$, unramified in $K/\mathbb{Q}$, such that for no place $\mathfrak{p}$ above $p$ the map $\tau^{\mathfrak p}$ is surjective. This is the general form of the theorem that the source proves probabilistically under its heuristic on the probability that $\tau^{\mathfrak p}$ is surjective. Numerical computations support the conjecture, but the general statement remains open.