The Schur $Q$-functions form a basis of the algebra $\Omega$ of symmetric functions generated by the odd-degree power sum basis $p_{d}$, and have ramifications in the projective representations of the symmetric group. So, as with ordinary Schur functions, it is relevant to consider the equality of skew Schur $Q$-functions $Q_{\lambda/\mu}$. This has been studied in 2008 by Barekat and van Willigenburg in the case when the shifted skew shape $\lambda/\mu$ is a ribbon. Building on this premise, we examine the case of near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture with evidence that all Schur $Q$ functions for frayed ribbon shapes are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the lattice walks version of the shifted Littlewood-Richardson rule, discovered in 2018 by Gillespie, Levinson, and Purbhoo.