The $3$-adic valuation of the cyclotomic product flips parity with $v_3(p+1)$

For $p \equiv 2 \pmod 3$, write $p = 3k+2$. Then $p^2-p+1 = 9k^2+9k+3$ carries exactly one factor of $3$, while $p+1 = 3(k+1)$ carries $v_3(p+1)$, and $3 \nmid p^2+p+1$. Hence

So the product has ODD $3$-adic valuation exactly when $v_3(p+1)$ is EVEN. Verified numerically for all $1052$ primes $p<40000$ with $p\equiv1\pmod4$ and $p\equiv2\pmod3$, with no counterexample.

Consequence for the $k=5$ two-prime residual. The first Dris equation reads $m^2 = (p^2+p+1)\cdot\tfrac{p+1}{2}(p^2-p+1)\cdot d_1^2 qr$, so the parity of that $3$-adic valuation must match the number of kernel primes equal to $3$. Together with three_kernel_prime_when_p_one_mod_three this makes the two cases complementary: every $p \equiv 1 \pmod 3$ forces $3$ into the kernel, and for $p \equiv 2 \pmod 3$ it does so exactly when $v_3(p+1)$ is even.