Smooth Gaussian-prime variance conjecture

Let $f$ be the smooth angular test function and let $\Phi$ be the smooth norm cutoff used to define the smoothed prime-angle count $\psi_{K,X}$. Write

$$c_2(f,\Phi)=\int_{-\infty}^{\infty}f(y)^2\,dy\int_0^{\infty}\Phi(t)^2\,dt.$$

Smooth Gaussian-prime variance conjecture. The variance satisfies

$$\operatorname{Var}(\psi_{K,X})\sim c_2(f,\Phi)\frac{X}{K}\min(\log X,2\log K).$$

The conjecture is formulated for the smoothed count and agrees with the elementary variance computation in the trivial regime $K\gg X$. A sharp-cutoff version is obtained heuristically by taking indicator functions for $f$ and $\Phi$, replacing the von Mangoldt weight by $\log X$, and ignoring higher prime powers.