Prime-divisor conjecture for the double-discriminant constant

Let $c_{n,k}$ denote the outlying constant, equivalently the greatest common divisor of the coefficients of $DD_{n,k}$, and let $\nu_p$ be the $p$-adic valuation. The conjecture concerns its prime divisors and valuations.

Double-discriminant constant conjecture. The following hold:

1. For $k>0$, $c_{n,k}$ has the same prime divisors as $2\gcd(k,n)$, while

$$c_{n,0}=2^{4\left\lfloor (n-1)/2\right\rfloor}.$$

2. $\nu_2(c_{n,k})\geq 4\left\lfloor\frac{n-1}{2}\right\rfloor$, with equality when $k=0$.
3. For all $n$, $k$, and $p$, $\nu_p(c_{n,k})$ is a multiple of $2p$.

These patterns are inferred from computations and compressed-polynomial experiments; for $n>6$, the tabulated constants are explicitly described as uncertain upper bounds. Thus the proposed complete description remains open.