Existence of graphs attaining neither general unbiased bound

Determine whether there exists a graph H satisfying 2\delta(H)-1\neq s_{MB}^1(H)\neq 2\Delta_c(H)-1.

Background

For unbiased Maker–Breaker games, the paper gives the lower bound 2\delta(H)-1 and an upper bound involving the maximum degree in a color class, 2\Delta_c(H)-1. Exact values are obtained for several families, but it remains unresolved whether some graph has a value strictly between these two general bounds.