I do not have a satisfactory answer to your question, just a pointer. In the paper:

Białas-Cież, Leokadia Markov sets in ${\bf C}$ are not polar. Bull. Polish Acad. Sci. Math. 46 (1998), no. 1, 83–89

for a compact subset $E$ of $\mathbb{C}$ which satisfies Markov inequality (i.e., certain estimate for derivatives of polynomials) a lower bound is proved for the capacity of $E$ in terms of the diameter of $E$ raised to the power 1/3. This is a global estimate, not a local one that you are asking about, but there may be a relation. The author of the paper may know more, so I suggest asking her anyway, whether you get any more specific answers on MO or not.

I do not have a satisfactory answer to your question, just a pointer. In the paper:

Białas-Cież, Leokadia Markov sets in ${\bf C}$ are not polar. Bull. Polish Acad. Sci. Math. 46 (1998), no. 1, 83–89

for a compact subset $E$ of $\mathbb{C}$ which satisfies Markov inequality (i.e., certain estimate for derivatives of polynomials) a lower bound is proved for the capacity of $E$ in terms of the diameter of $E$ raised to the power 1/3. This is a global estimate, not a local one that you are asking about, but there may be a relation. The author of the paper may know more, so I suggest asking her anyway, whether you get any more specific answers on MO or not.

BTW, the question could benefit from having an additional tag "Potential theory".