The alpha-generalized Erdős–Gallai path conjecture

Let $G$ be a 2-connected graph on $n$ vertices, and let $x,y\in V(G)$. Let $0<\alpha\leq\frac{1}{2}$. Alpha-generalized path conjecture. If $G-\{x,y\}$ contains more than $\alpha(n-2)$ vertices of degree at least $k$, then $G$ contains an $(x,y)$-path of length at least

$$2\alpha k.$$

This is proposed as a generalization of the paper's strengthened Erdős–Gallai theorem, recovering that theorem when $\alpha=\frac{1}{2}$. The paper offers it as a suggested conjecture and gives no resolution.