Asymmetric companion-zero conjecture for the smooth divisor-sum indicator

For $x>0$ and steepness parameter $\kappa>0$, define the smooth divisor-sum indicator by

$$\mathcal P_{\sigma}(x;\kappa)=\left(\sum_{i=2}^{\infty}\phi_{\kappa}\!\left(\frac{i}{x+1}\right)\frac{F(x,i)}{i}\right)-x.$$

For each odd prime $p\ge 3$, the asymmetric companion-zero conjecture. For sufficiently large $\kappa$, $\mathcal P_{\sigma}(\cdot;\kappa)$ has an asymmetric pair of real zeros flanking $p$: a left zero $x_p^-(\kappa)$ satisfying

$$p-x_p^-(\kappa)\asymp p\,\mathrm e^{-2\kappa/(p+1)}\qquad(\kappa\to\infty),$$

and a right zero whose distance from $p$ remains bounded away from zero as $\kappa\to\infty$. Numerics also indicate

$$\mathcal P_{\sigma}(p;\kappa)\approx-p\,\mathrm e^{-2\kappa/(p+1)}.$$

The claimed asymmetry contrasts with the two zeros approaching $p$ for $\mathcal P_\tau$; the limiting integer values of $\mathcal P_\sigma$ vanish precisely at primes, while the asserted right companion does not approach $p$.