Just to be clear when I said no, I am saying that I believe that is compatible with Romans chapter 7 talking about struggling Christians.
I must admit though that although I said no, it could be just the way I am looking at the matter.
Let me explain.

I believe that as Christians we have been implanted (if I can use that term) a new principle of life that has been planted by the Holy Spirit. The person becomes a new man and is being transformed into the image of Christ. They are of course as William Henrdriksen says in his commentary on Romans.
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Paul sees a different law, one that is constantly waging war against the law of his mind and making him a prisoner of the law of sin.
Verse 24 says this very forcefully "Wretched man that I am! Who will rescue me from this body of death?" I believe Paul here speaks for every true Christian.
Verse 25 shows that when a believer dies he will no longer be subject to the law of sin and death and he can serve and love the Lord the way he truly wants to.

So, in one sense as Christians we are still are affected by our fallen natures. However, being as we are definitely being transformed into the image of Christ in the here and now. Can we be considered "totally depraved" in the same sense that a non-Christian is?
I personally don't think so, but that might be because of how I am looking at the matter.