Persistence in probability for stochastic delay equations with multiplicative noise

We study the stochastic delay equation \begin{equation*} dX(t)=[-γX(t)+r f(X(t-τ))]\,dt+σX(t)\,dW(t), \end{equation*} where $γ,r,τ\in(0,\infty)$, $σ\in[0,\infty)$, and with $f$ either globally Lipschitz or globally bounded and locally Lipschitz, with $f(0)=0$ and $f(x)\ge0$ for all $x>0$. Under