Midway through the analysis, the argon ran out, and error code 11011: Vacuum (IF/BK) time out, vacuum too high to start turbo pump, appeared. I've attempted a full system restart, but the same error persists

Ok, I'm cleaning the cones, changing the graphite gasket and I will replace the oil on the pump as well to see if it works.

Larger are the orifices of the cones, particularly the sample cone, more chance you have to get the IF/Bk pressure errors. Usually when the sample cone is too used then is passing too much air or whatever trough it at the interface, and the ICP senses a pressure much higher than expected, and will not start the turbo. Better use a new Sample cone. Yes, the oil in the preliminary pump may help, but may not be the cause.

1101 Error: Vacuum (BK) time out, vacuum too high to start turbo pump.
The turbomolecular pump cannot work because the vacuum pressure (IF/BK) is not
below the limit within the prescribed time (When the instrument is changing from
Shutdown mode to Standby mode).
Check the vacuum pressure (IF/BK). The following causes are inferred.
• The vent valve is not closed or the seal is incomplete.
• The cover of the vacuum chamber is not closed or the seal is incomplete.
• The power of the foreline pump is not connected or the breaker is OFF.
• The foreline pump oil level is low or the oil is dirty.
• There is a leak in the gate valve.
• The vacuum tubing has failed (the tubing is off, or a leak is caused by an incomplete seal such
as a pin hole or a defective O-ring).
• There is a fault in the Pirani gauge (IF/BK) or the Connection PCA.
• There is a leak in the backing line, between turbo pump backing port to foreline pump inlet.

Hello,

Have you been able to resolve this? 

The cones do not affect the pressure required to get the instrument turbo pump to come on and get to standby mode.  They only become an issue when lighting plasma and when the interface valve opens.  

Most important thing is to make sure you have Argon pressure as this closes/opens the gate valve.  

Have the meter for the IF/BK pressure visible when you turn on the vacuum.  It needs to get below 2E1 for it to start the turbo.  If not there could be a leak somewhere.  Check the lid to the mass analyzer is positioned and push down to see if the pressure drops.  Check the vacuum hose for leaks, run your hand along it and see if you can feel air pulling into the hose.  Issue could be related to a faulty backing pump, or a fault in the Pirani gauge or it's circuitry.  If you can hear the pump turn on, and the IF/BK pressure never moves it is the gauge or the IF/BK valve not opening.  These will require a service engineer to replace.