Prime-polynomial symmetry conjecture for weighted Dirichlet polynomials

Let $P_f(s,X)$ be the weighted Dirichlet polynomial

$$P_f(s,X)= \sum_{p\leq X^2}\frac{v_{f,1}(e^{\log p/\log X})}{p^s},$$

where the sum is over primes, and let $f$ be a nonnegative mass-one $C^1([0,1])$-function supported in $[0,1]$. For a real number $\sigma$ and sufficiently large $T>0$, consider the maxima over $14\leq t\leq T$ and $3\leq X\leq t$.

Prime-polynomial symmetry conjecture. The positive and negative real parts, the real and imaginary parts, and the positive and negative imaginary parts have comparable maximal sizes:

$$\max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Re}(P_f(\sigma+it,X)) \asymp \max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Re}(-P_f(\sigma+it,X)),$$ $$\max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Re}(P_f(\sigma+it,X)) \asymp \max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Im}(P_f(\sigma+it,X)),$$

and

$$\max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Im}(P_f(\sigma+it,X)) \asymp \max_{14\leq t\leq T}\max_{3\leq X\leq t}\operatorname{Im}(-P_f(\sigma+it,X)).$$

The conjecture reflects the expected random distribution of the fractional parts $\{t\log p\}$ and would explain why the real and imaginary parts, together with their positive and negative parts, should have comparable maximal orders. Its status is not resolved in the supplied source.