Juniper supports should have been removed at planting.

The lollipop styled pear should be untethered now, if its rootball seems stable. Otherwise, wait until leaf drop this coming fall.

I would leave the stakes, but loosen up the straps so the trunk can move 2-3 inches. If no problems after 3 months, loosen more to about 6" of freedom of movement. Revaluate after 3-6 months and if good remove the stakes or loosen straps some more.

OK, I will look at loosening the stakes on the ornamental pear trees. I found a web page at the link below that says Juniper Skyrockets should not need stakes at all, so I will probably remove the stakes soon.

I noticed the link says the Skyrockets should have full sun. It does have pretty much full sun now, but I have planted ornamental pear trees on each side of the patio (you can see the one on the east side of the patio in the picture above) so that the patio will have some shade in the future for hot summer days.
The JS should outrun the pear trees in growth, but when the pear catches up starts providing shade, will the shade cause a problem for the junipers at that point?

Here is a link that might be useful: http://www.fast-growing-trees.com/JuniperSkyrocket.htm

"I found a web page at the link below that says Juniper Skyrockets should not need stakes at all..."

Most trees don't need stakes when planted. There's nothing special about the a Skyrocket Juniper. There are valid reasons to stake trees, and one of them is when you are working with a tree that has been unnaturally limbed up so that it looks like a lollipop.

It's normally not a bad idea to gradually loosen up the ties on a staked tree (to allow gradually increasing amounts of sway), but, with the stakes as close as they are to your pear tree's trunk, you won't be able to do much of that without potential for harming the trunk when it's windy (the tree could rub against the stake on the leeward side).

So, can the stakes on the ornamental pear trees be repositioned to allow more movement or should the stay the way they are for another year and then just remove the stakes completely?

When they are mature, they are supposed to look something like this photo.

Check the rootball by moving the trunk back and forth and see what moves. If the rootball is not pretty solid, loosen the ties as much as you can without leaving danger of the tree contacting the stakes in windy conditions. Then, after leaf drop this coming fall, remove the ties and the stakes. I wouldn't go the trouble of moving the stakes, and you may no longer need them anyway.