The constancy conjecture for the sieve functions $\Omega_k$ and $\Omega_k^{\mathrm{ext}}$

Let $\Omega_k(\vartheta,\vartheta_0)$ and $\Omega_k^{\mathrm{ext}}(\vartheta,\vartheta_0)$ be the functions defined in the source's sieve construction.

Constancy conjecture for $\Omega_k$ and $\Omega_k^{\mathrm{ext}}$. The function $\Omega_k(\vartheta,\vartheta_0)$ is constant with respect to $\vartheta_0$. The same applies to $\Omega_k^{\mathrm{ext}}$.

This proposed invariance concerns the auxiliary parameter $\vartheta_0$ in the GEH-case analysis. The supplied text gives no proof or resolution.