Conjecture that the critical group is not determined by the Tutte polynomial

For a connected undirected graph $G$, let $T(G;x,y)$ denote its Tutte polynomial and let $\EuScript{K}(G)$ denote its critical group. Non-determination conjecture. There exist connected undirected graphs $G$ and $H$ such that

$$T(G;x,y)=T(H;x,y)$$

and

$$\EuScript{K}(G)\not\simeq\EuScript{K}(H).$$

Thus the critical group cannot be computed from the Tutte polynomial alone. The source notes that the converse relation already fails: the Tutte polynomial is not computable from the critical group, and it proposes the reverse failure as an open conjecture.