Surviving Holy Roman Empire, Austro-Prussian Dualism

Suppose that for whatever reason, Francis II does not dissolve the Holy Roman Empire, and the crown is restored to the Habsburgs at the end of the Napoleonic Wars (if it is lost at all), along with the small states of the Rhine being restored under Imperial authority.

How does this alter Austro-Prussian relations? As I see it, they are the only two major German states, and I suspect Prussia would vie with Austria over supremacy. Would Prussia seek to dissolve the Empire, just as they had dissolved the German Confederation? Would they seek to take the crown from the Austrians?

I don't think a surviving HRE in itself would change that much initially, provided the internal reorganization still happens roughly as OTL, but it would have the long-term effect of making it somewhat more difficult to unify Germany without Austria. After all, the Habsburgs were basically seen as the legal rulers of the HRE, and by extension Germany, if only on paper (Quite different from the near hollow position they held at the top of the German Confederation, if only due to tradition).

Prussia was probably one of the biggest supporters of a continuing Empire and the concept of Austro-Prussian dualism in the 19th century. Frederick William IV was a romanticist with a fondness for the medieval HRE, and generally didn't oppose Austria. In fact, outright Austro-Prussian hostility really only started in the second half of the 19th century, and relations were generally cordial - but not perfect - before then.

With a PoD that allows the survival of the HRE, there's no reason to assume that Prussia will inevitably do what it did OTL. Prussia wasn't destined to dissolve the existing German framework and steal Austria's supremacy, and it could easily be avoided with a different Prussian leadership.