Shuffle One Variable Within Group

This question is an extension of the excellent answer provided by Robert Picard here: How to Randomly Assign to Groups of Different Sizes

We have this dataset, which is the same as in the previous question, but adds the year variable:

sysuse census, clear
keep state region pop
order state pop region
decode region, gen(reg)
replace reg="NCntrl" if reg=="N Cntrl"
drop region
gen year=20 
replace year=30 if _n>15
replace year=40 if _n>35

If I just wanted to re-randomly assign reg's across all observations (without regard to group), I could implement the answer to the previous post:

tempfile orig
save `orig'
keep reg
rename reg reg_new
set seed 234
gen double u = runiform()
sort u reg_new
merge 1:1 _n  using `orig', nogen

How would the code be modified so that reg is shuffled, but only within year? For example, there are 15 observations where year==20. These observations should be shuffled separately than the other years.