June schedules was updated early this week, removing all F seats (that were scheduled on CX841/4 second half of June). I would expect this trend to continue.

I suspect we have seen the end of First at CX.

Business Class far exceeds First Class standards of a decade ago.

I think due to low passenger demand, CX is deliberately not selling F tickets so they can easily swop flights from 77W to 359 or 35K without having to accommodate those who bought F tickets. I think F will be back either later this year or early next year.

First will definitely be back. As others have said far cheaper to run the 359s than the 77Ws. So only when sufficient demand is back will they consider. Today the demand is only low single percentages of what it used to be. No sense in flying the more expensive planes.

Agreed.
And CX cannot survive without positioning itself as a top premium airline.

Not when it comes to the number and quality of fellow passengers it doesn't.
First is about exclusivity as well as service levels.

Number, yes. Quality, depends on the type of pax you meet, if you understand what I mean.

It looks like CX is using a F equipped aircraft from MXP, but not selling F. Does anyone know if DM can select a F seat if ticketed in J?

Not sure which date you are talking about, so I found the next available flight, which is on 02AUG. There's no availability in F, but F class is there. It is not the case where they are selling the F cabin as J.

Any prediction will depend on the opening of travel.

CX has been postponing and postponing the resumption of travel to/from most European cities. Flights keep getting cancelled a month or so before.
Even when they schedule a month before flights to Europe have been massively cancelled.
They post that they resumed flying to FRA, but July only shows one flight every two weeks or so. They codeshare on LH.
They keep postponing the resumption to other major European cities. Right now LHR has a single daily flight in July (except for 1-3 July). AMS had resumed with 3 weekly, b ut it seems to get down to one weekly in July.

Whether MXP will resume soon is a big question mark.

CX is using A351 on all its European flights. They will start flying their 77W only when traffic has reached a much higher level where the "small" planes do not suffice.

All that to say that the question asked by the OP is highly theoretical.

Bumping this up. How does CX manage these type of flights when they use F-equipped 77Ws that they don't really "sell"?

I am pretty sure that 77W flight from Milan to HK being referred to (CX234) is the same aircraft originating from Pittsburgh (CX8802) as one of those seasonal cargo only passenger flights that does HKG-SGN-HKG-PIT-MXP-HKG. Last year it was FRA instead of MXP