Yuan–Zhang conjecture on exact Turán numbers of vertex-disjoint paths

Let $k_1\geq k_2\geq\ldots\geq k_m\geq 3$ with $k_1>3$, and let

$$F_m=P_{k_1}\cup P_{k_2}\cup\ldots\cup P_{k_m}$$

be the disjoint union of paths. For integers $n\geq m\geq 3$, write $c(n,a,b)$ and $c(n,a)$ as in the preceding definition. Yuan–Zhang's conjecture. The Turán number of $F_m$ is

$$\operatorname{ex}(n,F_m)=\max\left\{c(n,k_1,k_1),c(n,k_1+k_2,k_2),\ldots,c\left(n,\sum_{i=1}^{m}k_i,k_m\right),c\left(n,\sum_{i=1}^{m}\left\lfloor k_i/2\right\rfloor\right)+c\right\},$$

where $c=1$ if all of $k_1,k_2,\ldots,k_m$ are odd, and $c=0$ otherwise. This conjecture seeks the exact Turán number for every order $n$ of a graph excluding a prescribed union of vertex-disjoint paths, extending known results for matchings and for sufficiently large $n$. Its resolution is not established in the supplied source context.