I stumbled onto this talk on applications of rough paths to mathematical finance by Professor John Armstrong.
On slides 19-25 he presents a fascinating example of a sequence of rough paths that converge to 0 on the trace/path level, but has a nonzero signature given by the identity matrix times $t-s$.
The example is given without proof, but is intuitive. $X^n$ is defined as dissecting the interval $[0,1]$ and connecting $n$ Brownian bridges. By definition of the bridge process, it starts and ends at $0$. It is intuitive that if we join infinitely many, the resulting process must be identically $0$. I assume the iterated integrals are simply defined by using the fact that a bridge is an Ito diffusion, and some portion survives the limit.
This is quite an interesting example, because I believe the Brownian bridge is the solution of an SDE with additive noise (so I suppose Ito/Stratonovich coincide), so is geometric, but this limit is clearly a non geometric rough path. This is much more surprising than the similar example seen in Hairer's book on the "pure area path" in Ex. 2.10.
Unfortunately these are just slides with no proofs/theorems, I would like to read a much more detailed analysis of this example, but I can't find anything in the major textbooks on rough paths.
One article where they study the stochastic Lévy area for Brownian bridges in relation to the one for Brownian motion is "BROWNIAN BRIDGE EXPANSIONS FOR LÉVY AREA APPROXIMATIONS AND PARTICULAR VALUES OF THE RIEMANN ZETA FUNCTION".
Using $B_{r}=W_{r}-\frac{r}{1/n}W_{1/n}$, we write
$$\int_{0}^{1/n} B_{r}dB_{r}=\int_{0}^{1/n} (W_{r}-\frac{r}{1/n}W_{1/n})dW_{r}-\frac{W_{1/n}}{1/n}\int_{0}^{1/n} (W_{r}-\frac{r}{1/n}W_{1/n})dr$$
$$=\int_{0}^{1/n} W_{r}dW_{r}-\frac{W_{1/n}}{1/n}\int_{0}^{1/n}rdW_{r}-\frac{W_{1/n}}{1/n}\int_{0}^{1/n} W_{r}dr+(\frac{W_{1/n}}{1/n})^{2}\frac{1}{2n^{2}}$$
$$=\int_{0}^{1/n} W_{r}dW_{r}-\frac{(W_{1/n})^{2}}{2}$$
$$=\frac{1}{2}(W_{1/n}^{2}-\frac{1}{n})-\frac{(W_{1/n})^{2}}{2}$$
$$=-\frac{1}{2n}.$$
Then since there are $n$ of them we study
$$=-1/2.$$
So by rescalling everything, we get that the rough path lift converges to
$$\mathbb{B}_{t,s}^{n}\to -(t-s)/2$$
as mentioned in the slides.