We will now find out
Possibly - but only after a long delay!
As I understand CAA and EASA directives, and current airlines reaction to them, the initial basis of ALL investigation of potential ash encounters starts with EITHER in-flight observation of ash in the air or hitting the aircraft (St Elmo's Fire, ...?) OR the strong possibility that the route DID pass through an ash cloud. Then, specific ground inspection will be done, starting with Mark 1 Eyeball examination of the aircraft for 'traces of ash'.
As has already been discussed much earlier in this thread, there's a stack of good evidence that a flight CAN pass through quite dense ash with NO in-flight indication AT ALL and nothing found visually post-flight (in particular, the NASA DC-8 incident in 2000).
Seems to me that the key problem that remains completely unaddressed is any 'backstop' testing so that even in the absence of any other evidence, ash in turbine hot section cooling systems WILL be detected and appropriate maintenance done, ideally before major damage occurs. (Don't know whether any ash removal from inside blades is even theoretically possible without tearing the whole engine apart!) Filters on bleeds from the engines provide a DEFINITE indicator of ash ingestion but as far as I know there is no organised plan to test them post-flight.
Remember that a lot of aircraft tend to fly reciprocal routes (A to B, then B to A, maybe several times in the same day). Ash encountered on multiple flights will obviously become a CUMULATIVE problem, especially if low levels of ash are unexpectedly present on a busy route and not detected by any of the current means!