Does the exponential diophantine equation

$$ \prod x_i^{e_{x_i}} + \prod y_i^{e_{y_i}} = \prod z_i^{e_{z_i}}$$

with $x_i, y_i,z_i$ coprime and $e_{x_i}>3,e_{y_i}>3,e_{z_i}>3$ have solutions?

Any solution will give a good $abc$ triple. After browsing the list of $abc$ triples I did not find a solution in the list. Any heuristics?

*Update* nonsense relating conic to the $abc$ triple $7^2+2^{17} 181^2=3^8 809^2$ deleted as pointed out by several commenters.

Ideas about searching for solutions?