Aragão–Marciano–Mendonça's minimum-degree conjecture for path–clique Ramsey goodness

For positive integers $n,m,N$, a non-negative integer $t$, a graph $G$ on $N$ vertices, and the path $P_n$ and clique $K_m$, write $G\rightarrow(P_n,K_m)$ when every red–blue edge-coloring of $G$ contains a red copy of $P_n$ or a blue copy of $K_m$. Assume

$$(t+1)(n-1)(m-1)+1\leq N\leq(t+2)(n-1)(m-1).$$

Aragão–Marciano–Mendonça's conjecture. If

$$\delta(G)\geq N-\left\lceil\frac{t+1}{t+2}\left\lceil\frac{N}{m-1}\right\rceil\right\rceil,$$

then $G\rightarrow(P_n,K_m)$. This would extend the known minimum-degree condition for path–clique Ramsey goodness; the source notes that the conjecture was previously proved for $m=3$, while the paper studies related tree cases and partial confirmations.