Towards a conjecture on degree conditions for Ramsey goodness of paths

Recently, Aragão, Marciano, and Mendonça [\emph{European J. Combin.}, 2025] conjectured that for any graph $G$ on $n$ vertices satisfying $(r-1)(t-1)k<n\le (r-1)(t-1)(k+1)$, the minimum degree condition $\delta (G)\ge n-\lceil \frac{k}{k+1}\lceil \frac{n}{r-1}\rceil \rceil$ guarantees that $G\to (K_r,P_t)$. In this paper, we prove their conjecture for the regime $k\ge t-3$. Because the parameter $k$ scales linearly with the host graph order $n$, our result establishes the asymptotic truth of the conjecture.