a set of conditions under which a conversion formula holds for $\int_{0}^{T}\left\langle Y_{t}\circ\mathrm{d}\mathbf{X}_{t}\right\rangle -\delta^{X}\left(  Y\right)$. This formula is reminiscent of the one obtained for the case of second-order rough paths, but there are interesting differences too. Most notably the second term in , $$\begin{aligned}
 \label{non commut}
\int_{\lbrack0,T]^{2}}\mathds{1}_{[0,t)}(s)\mathrm{tr}\left[ J_{t}^{\mathbf{X}}\left(  J_{s}^{\mathbf{X}}\right)  ^{-1}V(Y_{s})-V(Y_{t})\right] \,\mathrm{d}R(s,t),\end{aligned}$$ which exists for $2 \leq p < 3$ as a well-defined 2D Young-Stieltjes integral, can only be identified as an $L^{2}$-limit of a sequence of approximating sums. The difference between the two cases stems from the lack of complementary Young regularity of the integrand and $R.$ Interestingly the integrand, while being continuous on $\left[  0,T\right]^{2}$, is not Hölder bi-continuous and so we cannot even appeal to the relaxed criteria discussed in point (1) above. It is unknown at present whether the limit is interpretable as a 2D Young-Stieltjes integral. We discuss in detail two important corollaries of our result. The first is where $X$ is a fractional Brownian motion with $H$ in $\left( \frac{1}{4}, \frac{1}{3} \right]$, and the second is the case where the vector fields defining commute. In this latter case, we show that the second term (\[non commut\]) in the correction formula disappears and, as a special 