Please comment on your experience with Plantation shutters. Yes they look great,

You are dealing with two issues with plantation shutters, 1) Diffusion off the blades which will be infinitely variable over 180 degrees (you could spend a lot of time trying to dial that in alone) and 2) First reflections off of the vertical glass behind them which will not diffuse at all. Neither will absorb any sound which IMHO is  a much more important issue when dealing with first reflections.

 I think any control of the sound waves that you might obtain by varying the pitch of the blades will be lost because of the reflecting nature of the glass immediately behind them. Heavy toe in and/or heavy drapes which you can open close when listening to music would be a much better resolution.

Now if the plantation shutters/windows are immediately behind the speakers or the listening chair, that would be an entirely different situation and I think they might work fairly well in place of more audiophile type of treatments.

Shutters will do little to nothing to help I'm afraid. Heavy drapes is an option but if you have the chance attaching something such as ASC sound planks to the window can work wonders. You can use Velcro to attach them so they can be removed to open the window when not needed. I am sure other similar items would work as well. Also consider the Marigo labs tuning dots for a very low profile way to damp the window

Marigo dots will do very little/nothing to help with the window.....lows travel through them like they were open.....highs get reflected right back at ya.  My house has great walls of glass windows....I use drapes recommended by the decorator....they look nice but do some, but little to perfect the room acoustically. Sometimes you go for the look, sometimes for the sound.....rarely can they be combined.

I have a double window (two completely seperate isolated panes mounted with a small offset between them to reduce chances of resonance in the window) with shutters in between as well. The inner window then has sound planks (three on one pane, two and a SR ART resonator and HFTs on the other) and also the large (40mm) Marigo dots on each pane

Each part helps but the main conclusion is you can (and probably should) treat a window like any other wall but obviously the end result is not going to win any interior design prizes ...

In a prior installation I lived with acres of glass and light treatment with the SR ART system and while no where near as acoustically well controlled as my current set-up it was very listenable

ps there is a fairly long track record of Marigo dots working well on exposed glass -- it can’t hurt give Ron Heydrich a call he’s a great guy and may have some good ideaas