Let's avoid rushing into hasty conclusions, because evidence establishes a conditionality; it does not create the sensitivity equivalence between methods based on fixed points and KKT.
The equivalence exists intrinsically; the proof reveals it under specific assumptions.
The weak link is the implicit universality, because this equivalence only holds if the conditions of the proof are maintained.
For example, an unexpected capital control by the PBOC or liquidity shocks in the Shanghai oil market would call into question the validity of this theoretical equivalence.
The exit condition is that this equivalence is only relevant if the established parameters are strictly followed, which is rarely the case in real markets.