engine start problem

[A8 Owner]

Hi, well since it went into the garage a cpl of weeks go, i have had nothing but problems with my car.
Here is the latest one.
When car is warm and you stop her.
If you turn her off and back on after a cpl of mins she will start as normal.
If tho you stop her and try after 5mins, its a different thing. It turns over just wont catch for a very long time.

Left 1.30 hrs it it starts a bit easyer.

Start in the morning starts like normal

The garage had the car back. It did it the night i dropped it off with them.
They then started work on it next day. Guess what they couldnt get it to do it. they put laptop on it, no codes show up.

What they did find is tho, the battery had 13v on it at the back. up front only 8v So took it to a battery company and agreed there was a problem with battery. So now had that changed, Flys over. didnt cure problem tho, as first start up wouldnt catch for along time (10-15 sec's)

So now i took some video clips
first clip is the car been started after 20min from getting battery




second clip about 2hr after coming home



Now as all the problems since they had it, have been in the area they were working. Front Pass wheel side.

reading forums have it down to.........
coolant sensor
evap emmision control
air flow meter
crank position sensor
cam position sensor
o2 sensor
throttle sensor
and the garage one is injector dripping.

Your views please