Bounded-degree property with exceptionally low budget

Establish whether there exists a bounded-degree monotone graph property P such that, when the exposure time t is of the order of the hitting time for P, a successful (t,b)-strategy exists with b=o(t/log n).

Background

The authors conjecture that for many bounded-degree properties, the budget near the hitting time cannot be substantially smaller than the hitting time. They ask whether any bounded-degree monotone property provides a counterexample with budget smaller than t/log n by an asymptotic factor.

References

Does there exist a bounded-degree monotone property $\mathcal{P}$ for which, if~$t$ is of the order of the hitting time for $\mathcal{P}$, there exists a successful $(t,b)$-strategy for $\mathcal{P}$ with $b=o(t/\log n)$?

— Graph factors and powers of Hamilton cycles in the budget-constrained random graph process  (2502.20870 - Díaz et al., 28 Feb 2025) in Problem, Section 7 “Final remarks and open problems”