VS CLimb?

[ppppilot]

If you want to save fuel, select 0 Cost Index and let the FMS manage the speed.
It is very difficult to explain all the relationships between IAS, V/S and fuel.
This is a very good study in airbus document:
getting to grips with fuel economy

Found at page 5:
This time difference plot has the same characteristics for all Airbus aircraft,
with the best time being obtained at the highest climb speed and max climb Mach number. Note that although a slow climb speed gets the aircraft to cruise altitude earlier, this requires more acceleration to cruise speed and more cruise to a given distance, making it slower overall.
The fuel difference plot characteristics vary with aircraft type.
The A310, A321 and A330 show similar characteristics to the A300 with a best fuel climb speed of about 290 to 300 knots.
The A318, A319 and A320 show better fuel burn at the lower speed range
(260 to 280 knots)
The A340 shows better fuel burn at the higher speed range (310-330 knots).
The A310 and A340 are similar to the A300 in showing minimum fuel at a
max climb Mach number of 0.78. In fact 0.8 is better for the A340-500/600.
However the A320 family and A330 benefit from the lower Mach No of 0.76.

In fact the more important to the fms at 0 cost index is to maintain an specific IAS, not a V/S. If you leave the second segment of the slope lift/drag and goes behind green dot (best ratio) you will burn more fuel even with reduced thrust.
Here you have another airbus document
Tailwinds