Infinitude conjecture for prime exponents in values of $3^{3+6c'}+1$

Let $c'$ range over integers with $c'\not\equiv 3\pmod 7$, and consider the prime factorization of $3^{3+6c'}+1$.

Infinitude conjecture. There are infinitely many values of $c'\not\equiv 3\pmod 7$ such that every prime other than $2$ and $7$ occurring in the factorization of $3^{3+6c'}+1$ has exponent congruent to $1\pmod 3$.

Such values of $c'$ would produce the polynomial families used to construct further twists of the Klein quartic that are counterexamples to the Hasse principle. The source provides no resolution of this arithmetic conjecture.