A question regarding the strong Markov property

In our lecture on Brownian motion & stochastic calculus we proved:

If $X$ is a canonical RCLL process having the strong Markov property and $\tau$ is a stopping time with $\tau < + \infty, P-a.s$, then the process $\sideset{^\tau}{}X$ (started at $\tau$ ) has the weak Markov property.

After the proof there is a remark:

In particular, if $X$ is Feller, then so is $\sideset{^\tau}{}X$, and therefore $\sideset{^\tau}{}X$ is again a strong Markov process.

Now, my question is:

If $X$ is a strong Markov process, isn't $\sideset{^\tau}{}X$ a strong Markov process in any case? Or, to put it differently, is there a mistake in my proof (see below)?

Let $\sigma$ be a stopping time, let $f \geq 0$ be a measurable function on the state space, and let $\mathcal{G}_\sigma := \mathcal{F}_{\tau + \sigma}$. Then,

$E[f( \sideset{^\tau}{_{\sigma+h}}X) | \mathcal{G}_{\sigma} ] = E[f(X_{\tau + \sigma + h}) | \mathcal{F}_{\tau + \sigma}] = E[f(X_{\tau + \sigma + h}) | \sigma (X_{\tau + \sigma})] = E[f(\sideset{^\tau}{_{\sigma + h}}X) | \sigma (\sideset{^\tau}{_\sigma}X)],$

$P-a.s.$ on $\{ \tau + \sigma < + \infty \}$ (where the second equality follows from the strong markov property applied to $\tau + \sigma$). And since $\tau < + \infty, P-a.s$, the above holds even $P-a.s.$ on $\{ \sigma < + \infty \}$, which is exactly the strong Markov property for the process $\sideset{^\tau}{}X$.

Thanks for you help!

Regards, Si

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