In R: find the closest value within group_by excluding self comparisons

I would like to find the value in a column that is has the absolute lowest difference to the each row in the same column.

I've tried solutions from find value closest to x by group in dplyr and return index from a vector of the value closest to a given element

My code:

library(DescTools)
data %>% 
  select(river, dist, id) %>%
  group_by(river) %>% 
  mutate(NNdist = Closest(dist, dist))

For id = TYWI03 I would expect NNdist = 1690 and id = TAFF04 I would expect NNdist = 05, but the value returned is the reference value i.e. It's returning a from Closest(x, a).

data is

 river  dist id     NNdist
  <chr> <dbl> <chr>   <dbl>
1 Tywi     34 TYWI03     34
2 Tywi   1690 TYWI02   1690
3 Tywi   1747 TYWI01   1747
4 Taff   1607 TAFF05   1607
5 Taff   4341 TAFF04   4341
6 Taff  12357 TAFF03  12357
7 Taff  16111 TAFF02  16111
8 Taff  18124 TAFF01  18124