Conlon–Fox–Sudakov–Wei conjecture for path threshold Ramsey multiplicity

Let $M(H,n)$ be the minimum number of monochromatic copies of a graph $H$ in a red/blue coloring of $K_n$, and define the threshold Ramsey multiplicity by

$$m(H)=M(H,r(H)),$$

where $r(H)$ is the Ramsey number of $H$. Conlon–Fox–Sudakov–Wei's path conjecture. For sufficiently large $t$,

$$m(P_{2t+1})=\frac{t}{2}(2t)!\qquad\text{and}\qquad m(P_{2t})=\frac{(2t)!}{2}.$$

The conjectured values arise from the corresponding critical split colorings, which are extremal for the Ramsey numbers of paths. The paper's abstract states that its upper bounds disprove the path conjectures, so this claim is refuted.