Can the sugar *lactose* enter the mitochondria ?, is there a symport of lactose with protons to the mitochondria ?.

In Prokayotes like e.coli, there is a lactose/H+ symporter in the cytoplasmatic membarane:

 

 

Such a symporter, can NOT exist for the cytoplasmatic membrane of Eukaryotes, because they do not have a proton gradient between this membrane. But, theoretically this transport can happen in the mitochondrial inner membrane. Does it happen in real life ?. If it does, what would the addition of valinomycin (which will transport K+ into the mitochondrial matrix, destroying the membrane electric potential) do for this symporter ?, will it make it slower ?, faster ?, why ?.

Well, since lactose is a secondary compound made of glucose and galactose, this break up occurs first via lactase, so technically lactose won't make it to the mitochondria, it will be enzymatically digested in the SI and carbohydrate metabolism will proceed in that direction.