R Data.table. Need modify column using grouping and complex conditional filtering

Here is data sample:

zz <- "
id  Sub_Segment1    Sub_Segment2    Sub_Segment3    Sub_Segment4    Sub_Segment5
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
1   x   x1  r   y1  z1
2   y   x2  r   y2  z1
2   y   x2  r   y2  z1
2   y   x2  r   y2  z1
2   y   x2  r   y2  z1
2   y   x2  r   y2  z1
"

Data <- read.table(text=zz, header = TRUE)
setDT(Data)

If I apply my modification to the whole table it returns NA:

Data[(length(unique(Sub_Segment1[Sub_Segment1!=""]))==1),name:="test" , by=id ]

returns:

 id Sub_Segment1 Sub_Segment2 Sub_Segment3 Sub_Segment4 Sub_Segment5 name
 1:  1            x           x1            r           y1           z1   NA
 2:  1            x           x1            r           y1           z1   NA
 3:  1            x           x1            r           y1           z1   NA
 4:  1            x           x1            r           y1           z1   NA
 5:  1            x           x1            r           y1           z1   NA
 6:  1            x           x1            r           y1           z1   NA
 7:  1            x           x1            r           y1           z1   NA
 8:  2            y           x2            r           y2           z1   NA
 9:  2            y           x2            r           y2           z1   NA
10:  2            y           x2            r           y2           z1   NA
11:  2            y           x2            r           y2           z1   NA
12:  2            y           x2            r           y2           z1   NA

But if I take only one sample with constant value in subsegment it works:

new_data = Data[id ==1]
new_data[(length(unique(Sub_Segment1[Sub_Segment1!=""]))==1) & id==1,name:="test" , by=id ]

returns correct

   id Sub_Segment1 Sub_Segment2 Sub_Segment3 Sub_Segment4 Sub_Segment5 name
1:  1            x           x1            r           y1           z1 test
2:  1            x           x1            r           y1           z1 test
3:  1            x           x1            r           y1           z1 test
4:  1            x           x1            r           y1           z1 test
5:  1            x           x1            r           y1           z1 test
6:  1            x           x1            r           y1           z1 test
7:  1            x           x1            r           y1           z1 test

How should I modify code to apply my function to the whole dataset?