Hi and welcome to the forum.
According to
this article, the pH difference between intermembrane space and mitocondrial matrix is about 1. Intermembrane space pH is little below 7 and pH in the matrix is roughly 8. The
pKa value is about equilibrium --> you can convert that to probability value for single DNP to exist as it's conjugate base. Conjugate base of DNP in inner membrane has higher change to be protoned from the intermembrane space than from the matrix. So, from time to time, it would take a proton from intermembrane space (lower pH) and transport it to the matrix (higher pH), driven by chemiosmotic force. This disturpts the proton gradient, and production of ATP. In example salicylic acid works as ionophore and disturps the proton gradient in mitocondria, and produces heat, if taken in high extent. Salicylic acid has low pKa value, but it still works the same way. So there must be a reason why it can act as an ionophore, and I think that the reason is the pH difference between intermembrane space and the matrix. CCCP also acts as ionophore in innermembrane and has lower pKa than physiological pH is. CCCP pKa is about 6, so it will be in its conjugated form roughly 50% of time (little bit below. Maybe 40%). So it can transport protons more efficiently trough innermembrane. Infact it is very toxic substance.
To backup this idea a little, I just put here an equation of chemical potential in case of electrochemical proton gradient found from
here:
Δμ = -Fφ + 2.3RT * ΔpH
ΔpH = pH of intermembrane space - pH of the matrix = -1
From this I would say that it is thermodynamicaly possible to transfer protons trough innermembrane, regardless of what substance does it.
Just some thoughts. This is not my special field, but the OPs question got my interest. Some molesnacks for you from that