Companion-zero conjecture for the smooth divisor-count indicator

For a prime $p$, a companion zero at $p$ is a real zero $x\neq p$ of a smooth prime indicator with $|x-p|<1$. A family $x_p^{(\pm)}(\kappa)$ is a prime-zero approximant if $x_p^{(\pm)}(\kappa)\to p$ as $\kappa\to\infty$. For each odd prime $p\ge 3$, the companion-zero conjecture. There exist $\kappa_0(p)\ge 1$ and $\varepsilon_p>0$ such that, for all $\kappa\ge\kappa_0(p)$, $\mathcal P_\tau(\cdot;\kappa)$ has two simple real zeros, one in $(p-\varepsilon_p,p)$ and one in $(p,p+\varepsilon_p)$. Numerical computations suggest that both distances from $p$ are $O\bigl(\mathrm e^{-\kappa/(p+1)}\bigr)$ as $\kappa\to\infty$.