Are there infinitely many primes of the form $\hspace{1mm} {}^b a \pm {}^a b$, with integers $a,b\geq 2$?

Let ${}^b a$ denote the tetration of $a$ to height $b$ (e.g., ${}^3 3=3^{3^3}=3^{27}=7625597484987$).

I am interested in primes of the forms ${}^b a+{}^a b$ and ${}^b a-{}^a b$, where $a,b \geq 2$ are integers (I came across this problem while I was computing the first terms of the more general sequences of primes of the form ${}^b a \pm {}^d c$, with $a,b,c,d\geq 2$, for submission to the OEIS).

Now, the only two terms I found of the more restrictive forms ${}^b a \pm {}^a b$ are:
${}^3 2+{}^2 3=16+27=43$ and ${}^2 3-{}^3 2=27-16=11$
(trivially, to be an odd prime, $a$ and $b$ must have opposite parity).

Question. Are there any other primes of the form ${}^b a + {}^a b \hspace{1mm}$ or $\hspace{1mm} {}^b a - {}^a b$? If yes, are there infinitely many primes of either form?

Remark. At a naive heuristic level, it is not clear to me what to expect: it seems that there are no immediate obstructions to primality beyond parity, while the extremely rapid growth of tetration makes prime values increasingly unlikely.
Even partial results for either the sum or the difference would be of interest.