Checking the Constraint Qualification 

I assumed all three to be binding (holding with equality around the 4 min mark), which implies c1=0 and c2=0 and k=p1*0+p2*0=0. When I only require c1=0 and c2=0 (around the 8 min mark), I do not require that k=0.

Constantin Bürgi what does it mean when you say constraints are binding ?

Binding means holding with equality (that is c1=0 instead of c1>=0 or c1>0).