The conjecture that the sieve parameter satisfies $\widetilde{M}(k)=k$

Let $\widetilde{M}(k)$ denote the parameter in Theorem. Conjecture on $\widetilde{M}(k)$. One can take $\widetilde{M}(k)=k$ for all $k\geq 1$ in Theorem. This would improve the paper's conditional bounds and is motivated by a natural extension of Goldbach's conjecture; the source gives no resolution, so the assertion remains open.