Primes of the form $a^2+b^2+c^2$, $0<a<b<c$

Question

Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero?

---

By the three square theorem of Legendre a natural number can be written as a square sum of three natural numbers if and only if it isn't of the form $4^i(8j+7)$. where $i,j$ are natural numbers.

Due to an answer to Conjecture: Any sufficiently big sum of three squares can be written as a square sum of three different natural numbers greater than zero there is a conjecture by Jeffrey Shallit:

A number is a sum of 3 squares, but not a sum of 3 distinct nonzero squares, if and only if it is of the form $4^js$, where $j \ge 0$ and

$s \in$
{1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 25, 27, 33, 34, 37, 43, 51, 57, 58, 67, 73, 82, 85, 97, 99, 102, 123, 130, 163, 177, 187, 193, 267, 627, 697}

So, if the conjecture by Shallit is true, then all primes not of the form $8m+7$ and not belonging to $\{2, 3, 5, 11, 13, 17, 19, 37, 43, 67, 73, 97, 163, 193\}$ can be written as a square sum of three different non zero natural numbers.

Answer 1

Nothing seems be known. See OEIS/A125516. Contrast with OEIS/A085317.