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Let $P=(p_{ij})$ and $Q=(q_{ij})$ be the $n\times n$ transition matrices for the two respective Markov chains, where $n\ge2$. Your conditions imply the following:
\begin{gather}p_{nn}=q_{nn},\\ p_{nj}<q_{nj}\text{ if }1\le j\le n-1. \end{gather} Hence, $1=\sum_{j=1}^n p_{nj}<\sum_{j=1}^n q_{nj}=1$, which is a contradiction (which makes any conclusion whatsoever true).