How to extend values down an r dataframe when row conditions are met?

Note that there are solutions to other questions that may resolve this specific question, such as https://stackoverflow.com/questions/40040834/replace-na-with-previous-or-next-value-by-group-using-dplyr. However, this question isn't about replacing NA's, NA's are OK in this question in certain circumstances. This question addresses replacing all cells in a group in a dataframe that fall after the the first non-NA in that group, with that first non-NA value. When I researched this issue I didn't find solutions that fit because I only want to replace NA's in certain circumstances (NA's in a group that occur prior to the first non-NA in that group remain; and a group with all NA's and no non-NA in that group retain all their NA's).

Is there a method, with a preference for dplyr or data.table, for extending target values down an R dataframe range when specified row conditions are met in a row within a group? I vaguely remember an rleid function in data.table that may do the trick but I'm having trouble implementing. Either as a new column or by over-writing existing column "State" in my below example.

For example, if we start with the below example dataframe, I'd like to send the target value of 1 in each row to the end of each ID grouping, after the first occurrence of that target value of 1 in a group, and as better explained in the illustration underneath:

myDF <- data.frame(
  ID = c(1,1,1,1,2,2,2,3,3,3),
  State = c(NA,NA,1,NA,1,NA,NA,NA,NA,NA))

You can use fill:

library(tidyr)
myDF %>%
  group_by(ID) %>%
  fill(State, .direction = "down")
# A tibble: 10 × 2
# Groups:   ID [3]
      ID State
   <dbl> <dbl>
 1     1    NA
 2     1    NA
 3     1     1
 4     1     1
 5     2     1
 6     2     1
 7     2     1
 8     3    NA
 9     3    NA
10     3    NA