Is natural filtration of a left-continuous modification of a Poisson process is right-continuous?

Question

I know that a natural filtration of a Poisson process is right-continuous. But does the same hold for a left-continuous modification of that Poisson process (denote $N'$). My guess, it does not hold. Since at the time of the first jump (it is totally inaccessible s.t.) $t_0$ we have $N'_{t_0}=0,$ so $\mathcal{F}_{t_0}=\{\emptyset, \Omega\}$, but strictly immediately after $t_0<s$ we have $\mathcal{F}_{s}=\sigma(N'_u,u\le s ) \ne \mathcal{F}_{t_0}.$