Let $T_G(x,y)$ denote the Tutte polynomial of a graph. Of course we may have $T_G(x,y) = T_H(x,y)$ for $G$ and $H$ non-isomorphic graphs.

Now let $c(G)$ denote the *cone graph* of $G$, i.e., the graph obtained from $G$ by adding a new vertex connected by an edge to every vertex of the original graph $G$. Let $c^{n}(G)$ denote the graph obtained by coning over $G$ $n$ times.

Note that we can have $T_G(x,y) = T_H(x,y)$ but $T_{c(G)}(x,y) \neq T_{c(H)}(x,y)$. For instance, take $G = K_{1,3}$ the star with three edges, and $H = P_3$ the path with three edges. Then $T_G(x,y) = T_H(x,y)$ because all trees on the same number of edges have the same Tutte polynomial, but $T_{c(G)}(x,y) \neq T_{c(H)}(x,y)$ because for instance $c(G)$ has $20$ spanning trees and $c(H)$ has $21$ spanning trees (and the number of spanning trees of any connected graph is the Tutte polynomial evaluated at $(1,1)$.)

**Question**: Is it the case that if $G$ and $H$ are non-isomorphic graphs then $T_{c^n(G)}(x,y) \neq T_{c^n(H)}(x,y)$ for some $n$? If so, are there some effective bounds on this $n$?