Conditional convergence of the Hadamard product

Question

The product $$\prod_\rho \left(1-\frac{s}{\rho}\right)$$ where $\rho$ ranges over all zeros of the Riemann zeta function is often referred to as the Hadamard product (because its convergence was first shown by Hadamard). It is often mentioned that the convergence is only conditional, and it can only be proven if we pair a zero $\rho$ with its symmetrical one $1-\rho$, so that the product actually becomes $$\prod_{\rm Im (\rho)\gt 0}\left(1-\frac{s(1-s)}{\rho(1-\rho}\right).$$ But I cannot find an actual proof that the convergence $\bf{is}$ conditional, that is, the product will diverge with a different order, or it will converge to different values. So I wonder if this is something unproven, or if there is a proof of it.