Theory
I am not familiar with the 777 fuel system. If anyone is in a position to share schematics or system descriptions I would be most interested. However, with the information gleaned from this and the previous thread plus the AAIB special report, I feel that the following scenario is plausible and deserves further investigation.
This appears to be a common-mode failure. The AAIB appears to have exonerated as far as reasonably possible the engine control systems. Another potential common-mode source seems to be the centre-tank fuel system. According to my reading of NSEU's post #187, the centre tank boost pumps are "override" AKA "priority" pumps i.e. they provide a higher delivery pressure than the wing tank pumps in order to use centre tank fuel first (wing bending relief). This theory rests on that premise.
1) Significant quantities of ice accumulate in the centre tank.
-Posssibly over more than one sector if it does not have chance to thaw during turn-arounds. Maybe supported by CT excessive water warnings reported in, I think, the previous thread, from the American "leak". Maybe exacerbated by scavenge systems contaminated by FOD. Added to during the final descent.
2) Ice melts during descent, hold, approach, yielding a few litres of free water in "empty" centre tank.
3) Config change to flaps 30 at ~1000' leads to attitude change allowing water to migrate to CT boost pump inlets.
Note by design, boost pumps are typically located at lowest point in tank at typical cruise attitude. Increased flap => more nose down for same speed.
4) Fault condition (or crew action) causes CT boost pumps to re-start, or fault condition allowed them to keep running when tank emptied with no indication (or no reported indication).
5) "Override" pump system design causes CT pumps to win over the wing tank pumps and water/fuel mix from CT is consequently supplied to engines in preferance to wing tank fuel.
Note that following the landing, the open LP (spar) valves allowed the contentents of the fuel feed pipes to be deposited onto the sod. Similarly, the CT fuel contents could not be assessed for water contamination as the tanks had been disrupted and subsequently contaminated by the firefighters.