(not the OP) I wouldn't say it is definitively true, but evidence to look for is data showing that STEM jobs in some states are paid relatively less than in other states after controlling for cost of living/average local wages: for instance, this graph[0] of wage variance for STEM occupations by state, along with this table[1] showing the wage difference for STEM vs other jobs in each state. (They don't show the exact relationship I'm after because neither of them has cost of living: STEM jobs in DC may only pay slightly higher than average but I assume that's because everyone is making a fair amount).
Cost of living isn't the only factor. Lifestyle quality is another, and not fully captured in the model you give because unlike money, people are non-homogenous. This may be more evident when the demand for a given field in a region is particularly small.
Consider [0] says that petroleum engineers are one of the highest paid STEM jobs, and [1] concurs. But a petroleum engineer in, say, North Dakota where there's an oil boom is likely going to have a higher salary than one in Hawaii even after adjusting for local wages. There are so few petroleum engineering jobs in Hawaii because it has no oil resources. If 0.1% of oil engineers are are willing to take a pay cut to move to Hawaii, then that small subpopulation may be enough to supply the Hawaii jobs market. (1 out of every 1000 jobs in ND is a petroleum engineer. We know that wasn't supplied by the indigenous population. While BLS doesn't have an entry for petroleum engineers in Hawaii, indicating that it's small.)
In a related vein, [1] points out the nuclear engineering is the highest paying job in Tennessee. This is probably influenced by Ridge National Labs and the three reactors in the state. The University of Tennessee offers a MS in nuclear engineering, so there's definitely a local production of people with those skills.
While Montana (no reactors, no nuclear research) doesn't have a BLS listing for nuclear engineers, which indicate that the demand for such is quite low. Low enough that secondary influences, like the desire to be near good fly fishing, or to move back to family in Montana, may be a bigger factor than cost-of-living.
Thus, regional salary variations, even when adjusted for CoL, may not give the estimate you're looking for.
As I read it, that proposes it's undesirable because there are few economic benefits for moving.
That would suggest that people are willing to move if there were stronger economic reasons, but the economic disparity across the US isn't high enough.
While the OP suggests that people aren't willing to move despite there being an economic disparity.
Thus, doesn't your link support the thesis the STEM shortage is a myth?
Is there actually a shortage of mobile qualified workers in the US? If so, what is your evidence?