WR,
I asked this question to a flight test engineer. He stated for line pilots we could consider approach speed (Vref +5) to be close enough to L/D speed.
He might not have wanted to recommend anything lower to
line pilots. I'm sure the laws of physics change depending on who's sat in the seat.
I don't know what the figures are - I would be interested to know. What I
am saying is that with all the LE & TE flap deployed (plus flaperon deflection), the wing profile becomes much more like the high-lift sections of old, where best performance occurs much closer to Vs. Given the amount of (deliberate) drag from the flap and the unavoidable bits from the gear, you may find that max. L/D occurs at a speed slower than 1.3/1.4Vs (or not... This is just a suggestion based on my experience with other aircraft.)
Also there was some thrust being produced by the engines: "above flight idle but below the commanded thrust" which complicates the situation somewhat. If you can get rid of a bit of drag by flying more slowly, that residual thrust will have more of an effect on the outcome. Food for thought...