Asymptotic cryptarithmic uniqueness of the prime sequence

Let the sequence of primes be ordered by increasing value, and call a prime cryptarithmically unique when no other prime has the same multiset of decimal digits. A sequence is asymptotically cryptarithmically unique if the proportion of its terms that are cryptarithmically unique tends to $1$. Asymptotic cryptarithmic-uniqueness conjecture. The sequence of primes is asymptotically cryptarithmically unique. The preceding estimates predict that the density of cryptarithmically unique primes eventually approaches unity, but the supplied source gives no proof of this asymptotic assertion.