Path-dependent rough differential equations in the full step-two range

We establish the existence of solutions to path-dependent rough differential equations with non-anticipative coefficients whose regularity is formulated in terms of horizontal and vertical Dupire derivatives. Under the same regularity assumptions as in Ananova~\cite{An23}, we develop a new causal Taylor argument and establish a second-order remainder estimate. This removes the regularity loss responsible for the restriction $2\le p<1+\sqrt{2}$ in~\cite{An23} and extends the existence result to the full range $2\le p<3$.