Under supermodularity conditions, the set of maximizers (argmax) of an objective function is known to be monotonically increasing with respect to multipliers or parameters of the function, as established in monotone comparative statics.
The retrieved papers include foundational and recent work in monotone comparative statics showing that supermodularity (and log-supermodularity) ensures that optimal solutions (such as argmax or equilibrium points) increase with parameters or multipliers. The evidence consistently supports the claim.