If anyone else ever needs it, it’s called a cam chain tensioner hose
1 question, is this an automatic chain tensioner that’s hydraulic? why doesn’t the second Gen have it?
as rob already wrote - it is an automatic working tensioneer - the gen2´s is totally different to gen1´s
the oil hose (#16) is not for getting oil pressure onto the tens. , but only oiling the tens. itself - suzuki changed it to this version in 2000 or 2001 - i guess.
but a little trick i have for the tens. BUT only for very experienced screwdrivers / mechanics
if the chain seems to be a very very little bit loose - you think you can hear a sound of the chain.
take out the screw (#12) put out the spring (#13) and the little shaft (#10)
then take an angled alen 3 or 4 wrench , put this with its longer side into and push this by hand a little so far / strong that a very little very quiet "CLICK" comes as the result.
reason - the spring´s force is not big enough to push the tens. into the next notch - AND thats absolutely normal.
this issue sometimes happens - but no need at all to replace any pieces.
and second hint
if you took out the screw allone you can "read" how much the chain had lengthen / how far the tens. went in
by the gap you find, when only holding the screw to the spring and messuring the gap "X" (
in red) shown at the very 1st pic
on my page
following pics (in 4 steps "Bild 1 - Bild 4") show approximately how many mm (with a "mm" slide gauge) the gap gets smaller when chain is lengthen.