The set of numbers of the form $q_1+q_2^2+q_3^3+q_4^4+q_5^5$ where all $q_k$ are primes.

Question

Conjecture:

All prime numbers greater than $109$ is of the form $\displaystyle\sum_{k=1}^5q_k^k$, where all $q_k$ are primes.

The conjecture is extracted from the question and answer here: Annoying primes

Tested for all primes $<70,000$.

The exceptional primes seems to be
$\{2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,109\}$

Are there heuristic arguments for the conjecture?

Also:
All primes except $\{2,3,5,7,11,13,17,19,23,29,37,43,53,61,67\}$ seems to be of the form $\displaystyle\sum_{k=1}^4q_k^k$.
All primes except $\{2,3,5,7,11,13,37,61,127\}$ seems to be of the form $\displaystyle\sum_{k=1}^3q_k^k$.

Both cases tested for primes less than $10,000$.

Answer 1

Here’s one small heuristic argument: there are no local obstructions, so lacking a good reason to the contrary, all large enough numbers should be of this form.

Another: There are $\gg x^{137/60-\varepsilon}$ representations up to $x$, and so each number has on average $x^{77/60-\varepsilon}$ representations. Since $$ \int \exp-x^{-76/60} $$ converges, the number of exceptions should be finite.