Conjecture on the Goldbach explicit-formula remainder under RH

Let $N$ be a positive integer, let $\rho_1$ and $\rho_2$ range over the nontrivial zeros of the Riemann zeta function, and let $B_x(a,b)$ denote the incomplete beta function. The notation $A\ll N$ means that $|A|$ is bounded by a constant multiple of $N$. Goldbach explicit-formula conjecture. Assuming the Riemann hypothesis, one has

$$\sum_{\rho_{1}}\left(2N\right)^{\rho_{1}}\left(\Gamma\left(\rho_{1}\right)\sum_{\rho_{2}}\frac{\left(2N\right)^{\rho_{2}}\Gamma\left(\rho_{2}\right)}{\Gamma\left(\rho_{1}+\rho_{2}+1\right)}-\sum_{\rho_{2}}\frac{\left(2N\right)^{\rho_{2}}}{\rho_{2}}\left(B_{1/N}\left(\rho_{2}+1,\rho_{1}\right)+B_{1/2}\left(\rho_{1},\rho_{2}+1\right)\right)\right)\ll N.$$

This estimate is proposed in connection with the explicit formula for Goldbach representations and would imply, together with suitable growth conditions, that sufficiently long intervals contain a Goldbach number. Its resolution is not supplied in the source.