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How much force does that spring apply? When I did mine the slack side was tight with the slightest amount of movement. When I posted my concern someone posted their's was similar so I just went with it. No problems so far.
Vagueness of terms... back run was tight with the slightest amount of movement, how much movement, how tight is tight. Suffice it to say when I did work with the cams doing the MCM the cams had to be pulled out to move a tooth, not enough play to move the chain without doing so. As long as there is some slack, the cam drive is good, providing it will turn over at least a few rotations by hand (ratchet/socket on left side crank bolt) after reinstalling the tensioner. Proves it is seated in correctly and nothing is holding it up or jamming in the sprockets. Turn counterclockwise, pulls on the front of the cam drive.
If using the OEM tensioner, pull the cap and remove the spring, pull back the ratchet pawl and push the plunger back in - then reinstall put the spring back in and put the cap bolt back on. If using a manual tensioner, back it out, install it (should not be pressing on the slider yet), then adjust it.