Can you carefully measure pins 1, 9, and 10 (ground, standby, and mute) to see what voltages are present on the TDA7296?  All three should be pulled low.  Without the schematic specific to the Rokit speakers, it's tough to say what the real purpose of those diodes are for.  It's almost like they're for a signal input to MUTE and STANDBY, but as far as I know, the Rokit speakers don't employ either function.  If they're pulled low, and remain low during power up and ON, then I wouldn't put much effort into replacing the diodes.

AFAIK the standby function is used (the speakers don't 'pop' when the power is applied), and there is a standby pin coming from the preamp board, which I have tested before - it is low at power-on and then quickly goes high.
 
I think i'm going to stick in some 1N4148s and see how I go. They definitely aren't power diodes or zeners. If the standby function is stopping the amp chips from powering on, that should be an easy problem to to diagnose.
 
Cheers

Upon further review of the datasheet, it appears that MUTE and STANDBY should be high to turn OFF their function.
 
I agree that they are not power/rectifier diodes or a zener, but I was questioning the latter myself, perhaps if they chose to create a reference voltage on the board for other components.  With the markings you saw to confirm, I think you'll be fine with replacement 1N4148's.

Success! This first board is working fine now :)

neilbags
Success! This first board is working fine now :)

Wonderful!  Congrats on a great fix!
 
Did you replace the black-gooped 1N4148's with new ones?  What else was fixed?  This information can definitely help others as well.  Not sure if you're also a Cakewalk user, but I'd also like to thank you for signing up and posting your question and troubles if it was perhaps just for your problems with the speakers.  I debate every now and then to re-create this thread over at Gearslutz, but I'm a) too lazy, and b) this thread brings the Cakewalk Forum a lot of attention.  It's a popular thread!   At some point I probably will as it's more fitting over there.  I started (under a different user name) another "ground-breaking" thread over at GS in regards to successfully repairing a non-working Alesis Andy A6.  It spawned a whole following of smart electronics dudes, such as yourself, to go above and beyond what I did and develop reprogrammed flash memory IC's, figure out boot code, and more!
 
 

2:43AM
Did you replace the black-gooped 1N4148's with new ones?  What else was fixed?

There were 4 sections of the board that were damaged. I removed all the components from the affected areas, cleaned them and the board and replaced any that looked suspect. Mostly resistors, a couple of jumpers and those 1N4148s. There were some small electrolytic caps with corrosion as well but after cleaning them up and testing them they seemed ok.

Glad I found this thread. I posted this on /r/audiorepairs:
 
I bought these (Rokit 6 G2) used about 2 months ago and it's probably have been a few days that I noticed my setup isn't balanced anymore. Turns out one of them now outputs quieter at the same volume/preamp setting. Sometimes it's unstable like suddenly going louder/quieter. If I crank the volume up the bass blows out / distorts. I've tried switching cables with the other one and it's still there. I already checked my computer and my output is balanced.
I can do soldering and minor repairs, should I open it up and see if any components fail? Any chance that the driver/cone itself needs to be replaced?
 
So I see that there's a huge problem with the build quality (especially the goop). Explains the hissing I hear since the day I got it. So I'm pretty proficient with a solder but I don't know stuff about working with capacitors and checking the components. Can someone link me a guide to the basics so I can start diagnosing my speakers? And should I buy the usual capacitors or audio grade capacitors to replace the components? Thanks a lot to 2:46AM for your detailed guide so far.

My bet's on the black goop. If you can solder, you should be able to do what I did and clean up the board. You will probably have damaged resistors.
 
I tested the main filter caps (the big ones) with a cheap ESR meter like this one:
https://www.banggood.com/LCR-T4-12864LCD-ESR-SCR-Meter-Transistor-Tester-p-978992.html?rmmds=search
 
But all my caps were OK. Be aware when opening this that the black goop could have damaged the bleeder resistors which are connected across the caps, so those capacitors could still be holding a significant charge. Make sure you have a way to discharge the caps before you go touching the board.

so I opened up my monitor and got these pictures (imgur.com/a/8t319). As far as I can see my caps are ok except the one in front of the TDA7296. How should I check the IC condition? I only have a multimeter on hand. Or should I just start cleaning up the black goop?

Man these speakers suck!
 
I would just start cleaning the goop.  IMO, don't worry about the amplifier IC.  When in operation and powered up, you risk seriously shorting out the component.