The discriminant conjecture for the field generated by a root of $f_{r,p}$

Let $p$ be the odd prime and let $\theta$ be the algebraic number considered in the preceding discriminant computation, with $\mathbb{Q}(\theta)$ its generated number field. Discriminant conjecture. The exact power of $2$ dividing the discriminant of $\mathbb{Q}(\theta)$ is

$$2^{\frac{3}{2}(p-1)}.$$

The preceding argument gives a discriminant divisible by the stated expression up to a square factor, while computations suggest that the defining polynomial has a larger power of $2$; the conjectured field-discriminant valuation is not proved in the supplied text.