I don't think any "conventional" organic solvent can dissolve inorganic precipitates. you can try some chelating agents (e.g., EDTA) to increase Mn solubility.
Synthesize MnO nanocrystals according to the attached paper. The nanocrystals have olic acid as surface ligands so that they are lipophilic and can be dissolved in nonpolar solvents such as hexane.
I used liposome to encapsulate MnO2, and the particles can be dissolved in cyclohexane. However, after centrifugation, washing with alcohol and changing them into chloroform, they became viscous (easily form aggregation) and hard to be dissolved. It could be possible that I use too less lipid to do the formation.
Did you use the method in that paper to prepare nanosized MnO2? Perhaps you can try to exchange the surface ligand of MnO2 (or simply remove the olic ligand) using method provided in this paper: J. Am. Chem. Soc. 2011, 133, 998–1006. I guess the aggregation is due to the polarity of chloroform.
Mn in MnO2 is in +4 oxidation state, making it insoluble in an acidic and alkaline medium (pH10) followed by oxidation using strong oxidants like persulfate/H2O2. Alternatively, Mn can also be precipitated as MnCO3, which in the temperature range 300-400oC can be reduced to nano-range MnO2