Fraisse's conjecture, now a theorem proved by Laver, states that the class of countable linear orders is well-quasi-ordered under embeddability. The reverse mathematics of this theorem is known to require significant logical strength, sitting at or near ATR0 for certain formulations, but the precise calibration of its proof-theoretic strength for natural restricted subclasses remains unresolved. Specifically, when attention is confined to linear orders of bounded or finite width as quasi-orders, it is unknown whether the resulting statement is strictly weaker than the full conjecture, and where exactly it lands in the hierarchy of subsystems of second-order arithmetic between ATR0 and Pi11-CA0.