Performing an SN1 reaction in a polar aprotic solvent Straightforward question
What would be the outcome of having an SN1 reaction carried out in a polar aprotic solvent? I have found this question approached scarcely and with unsatisfactory answers.
What I know so far
I know the SN1 reaction is carried out in a polar protic solvent since it helps stabilize the carbocation that gets formed (thanks to the lone pairs adjacent to the electronegative atoms of the solvent itself) and also the leaving group (stabilized by hydrogen bonding).

Also, many such reactions are performed as solvolysis and it is preferred - and is much more efficient - to have the solvent act as the nucleophile when this is possible. Based on this we can ascertain the fact that a polar protic solvent would be the best choice for such a reaction and its polar aprotic homologue would not do as well.

All of the reaction rates comparisons I have seen so far take into account a polar protic solvent and a nonpolar one. But what about the protic-aprotic dichotomy?
Final note
However, my question is - can we not use a polar aprotic solvent at all? I have no problem imagining it would perform in a less efficient manner. However, stabilization is not completely absent - at least it has the capacity to stabilize the carbocation. Would the sheer lack of stabilization of the leaving group (and of the nucleophile even though many SN1 reactions do not use charged nucleophiles) render this kind of solvent completely useless for this reaction? Or would it work to some extent?
The rate of an $S_N1$ reaction may be increased by increasing the polarity of the solvent of reaction.  Because, the more polar a solvent (as measured by the dielectric constant / relative permitivity, the easier to separate ions from each other; here: carbocation and its counter anion.
Contrary to polar-aprotic solvents (think DMF, DMSO) eventually leading to «naked ions», polar-protic solvents (like ethanol, methanol, water) may weakly interact with these ions; similar to a Lewis acid/base reaction, they stabilize the ions to stay in proximity to each other as an ion pair.  Eventually, it is by experiment to figure out which effect is dominating at the reaction conditions (structure of the substrate and reagents, concentration of starting material and reagents, temperature, etc.).