Fractional matching gap conjecture for critical graphs without 1-factors

Let $k\geq 3$ and let $G$ be a $k$-critical graph. Let $\mu(G)$ denote the matching number and $\mu_f(G)$ the fractional matching number. Fractional matching gap conjecture. If $G$ does not have a 1-factor, then

$$\mu_f(G)>\mu(G).$$

The source says this conjecture is unsolved even for critical graphs with a near-perfect matching, and that it would follow from the preceding fractional perfect matching conjecture.