Erdős–Faudree–Schelp–Simonovits conjecture on high-degree vertices and long paths

For positive integers $n>d\geq k$, let $\phi(n,d,k)$ be the smallest integer such that every $n$-vertex graph with at least $\phi(n,d,k)$ vertices of degree at least $d$ contains a path $P_{k+1}$ on $k+1$ vertices. Erdős–Faudree–Schelp–Simonovits conjecture. For any positive integers $n>d\geq k$,

$$\phi(n,d,k)\leq \left\lfloor \frac{k-1}{2}\right\rfloor\left\lfloor\frac{n}{d+1}\right\rfloor+\epsilon,$$

where $\epsilon=1$ if $k$ is odd and $\epsilon=2$ otherwise. The conjecture gives the proposed sharp general upper bound for the minimum number of high-degree vertices forcing a path on $k+1$ vertices; the paper presents a complete solution to this extremal problem, but the supplied material does not state whether this exact conjecture is resolved.