Conjecture: $xa^y+yb^z=zc^w+wd^x$ has no solutions in distinct positive primes satisfying $a + b + c + d = x + y + z + w$

Conjecture:

Does the Diophantine equation $$x \cdot a^y + y \cdot b^z = z \cdot c^w + w \cdot d^x$$

And satisfying $$a + b + c + d = x + y + z + w$$

Also $$a \neq b \neq c \neq d \neq x \neq y \neq z \neq w$$

have any solutions in primes?

Motivation:

In a previous iteration of this problem, allowing composite numbers yielded counterexamples, including an infinite parametric family of odd solutions using form constructions such as $T = \frac{r^{63}+r^{21}+r^9+r^7}{20}$. which fail when all 8 variables are strictly restricted to primes.

(About details, see this question)

Question:

Are there any prime solutions to this system?