Does $\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$ with $p_i$ distincts primes and $k\geq2$ have a solution ?

Here is what I already know :

There is no solutions if $k\equiv0\bmod2$ or if $k\equiv0\bmod3$.

It's a special case of Hurwitz equation ($A\prod_{i=1}^{k}p_i=\sum_{i=1}^{k}p_i^2$) with $A=1$ and a fondamental solution can't contain coprimes.

There is no solution for $k\leq12$.

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