Loadout question: grenades with two hands full

Can you throw a grenade if holding a musical instrument and a trench shield?

I was wondering if I could run this loadout in New Antioch Prussian combat engineer but I'm not sure if it's legal. The grenade keyword says you don't have to hold it but I can't find anywhere that says you switch weapons to grenades when you use them so I don't know that the rule on the shield and instrument saying they cannot be switched out or take one hand at all times respectively would apply, have I missed something?

Honestly don't know if it would even be good, but before I test it I wanted to be sure its valid.

The trench shield reads 'Always takes one hand to use in both melee and in ranged combat, and cannot be switched out.'

The musical instrument reads 'Musical Instruments take one hand to use at all times as if it were a weapon.'

The grenade keyword reads 'They do not count towards the number of ranged weapons a model can carry and do not have to be held in your hand at all times.'