Let $M$ be a matroid. (Assume that $M$ is a graphic matroid if it helps.) Let $M^2 = M \vee M$ be a union of $M$ by itself. See e.g. this lecture note for the definition of matroid union.
From the definition of matroid union, we have the following: There is a fixed partitioning $P$ of all independent sets of $M^2$ such that for any independent set $I$ of $M^2$, we have $P(I) = (I_1,I_2)$ where $I_1$ and $I_2$ are disjoint independent sets of $M$.
Is there a partition $P$ that also has the additional "robustness" property below?
For any two independent sets $I$ and $I'$ where the size of symmetric difference $|I \triangle I'|$ is small, say at most 2, we have both $|I_1 \triangle I'_1|$ and $|I_2 \triangle I'_2|$ are small, say at most some constant.