Given Oort's definition, if perigee is ≥5000 AU and the orbit isn't cleared, it's part of the Oort cloud.
There are multiple detected bodies matching that. Not as many as the Kuiper Belt, but it really only takes a handful to establish the pattern.
Actually, by the definition on phys.org, 2000 AU is sufficient.
2015-08-oort-cloudThe Oort Cloud is through to extend from between 2,000 and 5,000 AU (0.03 and 0.08 ly) to as far as 50,000 AU (0.79 ly) from the sun, though some estimates place the outer edge as far as 100,000 and 200,000 AU (1.58 and 3.16 ly). The Cloud is thought to be comprised of two regions – a spherical outer Oort Cloud of 20,000 – 50,000 AU (0.32 – 0.79 ly), and disc-shaped inner Oort (or Hills) Cloud of 2,000 – 20,000 AU (0.03 – 0.32 ly).
Read more at: https://phys.org/news/2015-08-oort-cloud.html#jCp
I don't know of any object with a perihelion of greater than 2000 AU - the most distant perihelion for a comet in wolfram is 11.78 au for 167P/CINEOS and the most distant perihelion for a minor planet is (note, edited: 80 au) for 2012 VP113. Wolfram is by no means exhaustive, however. I'm all but certain there are such objects, I just am unaware of them -
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